1712indexindex.xmlDeligne's "Hodge Theory I, II, and III"
This is an English translation of P. Deligne's three papers on Hodge theory:
"Théorie de Hodge I", Actes du Congrès intern. math. 1 (1970) pp. 425–430.
[PDF]
"Théorie de Hodge II", Pub. Math. de l'IHÉS 40 (1971) pp. 5–58.
[PDF]
"Théorie de Hodge III", Pub. Math. de l'IHÉS 44 (1974) pp. 5–77.
[PDF]TO-DO: replace fref by ref
The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
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Hodge Theory I ✓
Hodge Theory II
Introduction ✓
Filtrations ✓
Filtered objects
Opposite filtrations
The two filtrations lemma
Hypercohomology of filtered complexes
Hodge structures ✓
Pure structures
Hodge theory
Mixed structures
Hodge theory of non-singular algebraic varieties ✓
Logarithmic poles and residues
Mixed Hodge theory
Applications and supplements
The fixed set theorem ✓
The semi-simplicity theorem
Supplement to [D1968]
Homomorphisms from abelian schemes
Hodge Theory III
Introduction
Terminology and notation
Cohomological descent
Examples of simplicial topological spaces
Supplements to §1
Hodge theory of algebraic spaces
Examples and applications
Hodge theory in level \leqslant 1
Bibliography1691hodge-theory-ii-4hodge-theory-ii-4.xmlHodge Theory II › Applications and supplements4hodge-theory-ii1647hodge-theory-ii-4.1hodge-theory-ii-4.1.xmlThe fixed set theorem4.1hodge-theory-ii-41495Theoremhodge-theory-ii-4.1.1hodge-theory-ii-4.1.1.xml4.1.1hodge-theory-ii-4.1
Let S be a smooth separated scheme, and f \colon X \to S a smooth proper morphism.
In rational cohomology, the Leray spectral sequence
E_2^{pq} = \operatorname {H} ^p(S, \mathrm {R} ^qf_* \mathbb {Q} ) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {Q} )
degenerates (E_2=E_ \infty).
If \bar {X} is a non-singular compactification of X, then the canonical morphism
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^0(S, \mathrm {R} ^nf_* \mathbb {Q} )
is surjective.
266Proofhodge-theory-ii-4.1.1
For f smooth and projective, claim (i) is proven in D1968.
The proof in D1968 is rewritten in a more readable manner in G1970;
it does not use the smoothness of S.
Let us fix f, and prove that (i)\implies(ii).
We can reduce to the case where S is non-empty and connected.
If s \in S is a point, then the local system \mathrm {R} ^nf_* \mathbb {Q} is entirely described by its fibre ( \mathrm {R} ^nf_* \mathbb {Q} )_s at s and the action of the fundamental group \pi = \pi _1(S,s) on this fibre.
We have
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xrightarrow { \sim } [( \mathrm {R} ^n f_* \mathbb {Q} )_s]^ \pi .
If X_s=f^{-1}(s), then the composite arrow
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \to ( \mathrm {R} ^n f_* \mathbb {Q} ) \approx \operatorname {H} ^n(X_s, \mathbb {Q} )
is thus injective.
Consider the arrows
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \xrightarrow {a} \operatorname {H} ^n(X, \mathbb {Q} ) \xrightarrow {b} \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xhookrightarrow {c} \operatorname {H} ^n(X_s, \mathbb {Q} ).
The arrows cb and cba have the same image, by .
The arrow b is an "edge homomorphism" of the Leray spectral sequence, and is thus surjective by hypothesis.
Since c is injective, b and ba have the same image, and ba is surjective.
This proves (ii) for f projective.
From this we will deduce the general case.
We can again reduce to the case where S is non-empty and connected.
By Chow's lemma and the resolution of singularities, there exists a quasi-projective smooth scheme X' and a projective birational morphism p \colon X' \to X.
There thus exists (by Bertini, or Sard) a non-empty Zariski open S_1 of S such that X'_1=(fp)^{-1}(S_1) is smooth over S_1.
Finally, let X_1=f^{-1}(S_1), let \bar {X} be a smooth compactification of X, and let \bar {X}'_1 be a smooth compactification of X'_1 that dominates \bar {X}.
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
\bar {X}
&& \bar {X}'_1
\ar [ll,swap," \bar {p}"]
\\ X
\ar [u,hook]
\ar [d,swap,"f"]
& X_1
\ar [l,hook']
\ar [d,swap,"f_1"]
& X'_1
\ar [u,hook]
\ar [l,swap,"p"]
\ar [d,swap,"f'_1"]
\\ S
& S_1
\ar [l,hook',"i"]
& S_1
\ar [l,equals]
\end {tikzcd}
If s \in S_1, then the morphism i_* \colon \pi _1(S_1,s) \to \pi _1(S,s) is surjective, since the topological codimension of S \setminus S_1 is \geqslant2.
We thus have
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xrightarrow { \sim } \operatorname {H} ^0(S_1, \mathrm {R} ^n {f_1}_* \mathbb {Q} )
and it suffices to prove that
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^0(S_1, \mathrm {R} ^n {f_1}_* \mathbb {Q} )
is surjective.
The vertical arrows in the diagram
\begin {CD} \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) @>>> \operatorname {H} ^n(X_1, \mathbb {Q} ) @>>> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f_1}_* \mathbb {Q} ) \\ @VV{ \bar {p}^*}V @VV{p^*}V @VV{p^*}V \\ \operatorname {H} ^n( \bar {X}'_1, \mathbb {Q} ) @>>> \operatorname {H} ^n(X'_1, \mathbb {Q} ) @>>> \operatorname {H} ^0(S_1, \mathrm {R} {f'_1}_* \mathbb {Q} ) \end {CD}
admit left inverses given by the Gysin morphisms \bar {p}_!, p_!, and p_! (respectively), defined by Poincaré duality as the transposes of the arrows analogous to the above but in cohomology with proper support.
Furthermore, the diagram
\begin {CD} \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) @>{u}>> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f_1}_* \mathbb {Q} ) \\ @A{ \bar {p}_!}AA @AA{p_!}A \\ \operatorname {H} ^n( \bar {X}', \mathbb {Q} ) @>>{v}> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f'_1}_* \mathbb {Q} ) \end {CD}
commutes.
The arrow u is thus a direct factor of the arrow v.
Since v is surjective (because f'_1 is projective), so too is u.
We now prove (i) in the general case.
We can reduce to the case where f is of pure relative dimension n.
Let f \times f be the projection of X \times _S X to S.
Let \delta be the image of the cohomology class of the diagonal of X \times _S X in \operatorname {H} ^0(S, \mathrm {R} ^n(f \times f)_*, \mathbb {Q} ).
We have, by Künneth,
\mathrm {R} ^n(f \times f)_* \mathbb {Q} = \sum _{p+q=n} ( \mathrm {R} ^p f_* \mathbb {Q} \otimes \mathrm {R} ^q f_* \mathbb {Q} ).
We denote by \delta '_{pq} (for p+q=n) the components of \delta in this decomposition, and by \delta _{pq} the classes in \operatorname {H} ^n(X \times _S X) given by the images of the \delta '_{pq}.
The \delta _{pq} define, in the derived category D^+(S), homomorphisms
\delta _p \colon \mathrm {R} f_* \mathbb {Q} \to \mathrm {R} f_* \mathbb {Q}
such that \operatorname { \mathscr {H}} ^q( \delta _p) is zero for p \neq q, and is the identity for p=q.
By [D1968], we thus have
265Equationhodge-theory-ii-4.1.1.1hodge-theory-ii-4.1.1.1.xml4.1.1.1hodge-theory-ii-4.1.1 \mathrm {R} f_* \mathbb {Q} \approx \sum _p \mathrm {R} ^p f_* \mathbb {Q} [-p] \tag{4.1.1.1}
in D^+(S), and the Leray spectral sequence degenerates.
1496Corollaryhodge-theory-ii-4.1.2hodge-theory-ii-4.1.2.xml4.1.2hodge-theory-ii-4.1
Let f \colon X \to S be a proper smooth morphism, with S reduced, connected, and separated.
Let ( \mathrm {R} ^n f_* \mathbb {Q} )^0 be the largest constant local system on \mathrm {R} ^n f_* \mathbb {Q}, with fibre \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ).
Then, for each s \in S, ( \mathrm {R} ^n f_* \mathbb {Q} )_s^0 is a Hodge sub-structure of ( \mathrm {R} ^n f_* \mathbb {Q} ) \simeq \operatorname {H} ^n(X_s, \mathbb {Q} ), and the induced Hodge structure on \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) is independent of s.
1492Proofhodge-theory-ii-4.1.2
Let s \in S, and let X_s=f^{-1}(s).
If S is smooth, and if \bar {X} is a smooth compactification of X, then, by , the subspace ( \mathrm {R} ^n f_* \mathbb {Q} )_s^0 of ( \mathrm {R} ^n f_* \mathbb {Q} )_s is the image of \operatorname {H} ^n( \bar {X}, \mathbb {Q} ).
Since the restriction map
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^n(X_s, \mathbb {Q} )
is a morphism of Hodge structures, its image is a Hodge sub-structure, and the induced Hodge structure on \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ), given by the quotient of that on \operatorname {H} ^n( \bar {X}, \mathbb {Q} ), is independent of s.
In the general case, also implies that, if a is a global section of \mathrm {R} ^n f_* \mathbb {C}, then its components a^{p,q} of type (p,q), which are a priori only continuous sections of the complex bundle defined by \mathrm {R} ^n f_* \mathbb {C}, are in fact locally constant, i.e. are sections of \mathrm {R} ^n f_* \mathbb {C}.
This is the case because a^{p,q} is continuous, and further locally constant on the (dense) open subset of smooth points of S, by the above.
1497hodge-theory-ii-4.1.3hodge-theory-ii-4.1.3.xml4.1.3hodge-theory-ii-4.1
In the case where S is compact, a generalisation of is proven by analytic methods in [G1970].
As a corollary to , we note:
268hodge-theory-ii-4.1.3.1hodge-theory-ii-4.1.3.1.xml4.1.3.1hodge-theory-ii-4.1.3
(cf. [G1970]).
Let f \colon X \to S be a proper smooth morphism, as in .
If a global section a of \mathrm {R} ^n f_* \mathbb {C} is of Hodge type (p,q) at a point, then a is everywhere of type (p,q).
In particular, if n=2 and if a is, at some point s, the cohomology class of a divisor of X_s, then a is everywhere the cohomology class of a divisor, and, if S is smooth, is even defined by a divisor D on X.
269hodge-theory-ii-4.1.3.2hodge-theory-ii-4.1.3.2.xml4.1.3.2hodge-theory-ii-4.1.3
(cf. [G1966]).
Let S be a reduced connected scheme of finite type over \mathbb {C}, and let f_1 \colon X_1 \to S and f_2 \colon X_2 \to S be abelian schemes over S.
If a morphism u \colon \mathrm {R} ^1{f_2}_* \mathbb {Z} \to \mathrm {R} ^1{f_1}_* \mathbb {Z} comes from a morphism \widetilde {u}_s \colon (X_1)_s \to (X_2)_s of abelian varieties at some point s \in S, then u comes from a (unique) morphism of abelian schemes \widetilde {u} \colon X_1 \to X_2.
270hodge-theory-ii-4.1.3.3hodge-theory-ii-4.1.3.3.xml4.1.3.3hodge-theory-ii-4.1.3
(cf. [K1971]).
Let S be a smooth connected scheme, s \in S a point, f \colon X \to S a proper smooth morphism, and P a direct factor of the local system \mathrm {R} ^i f_* \mathbb {Q}, which is pointwise a Hodge sub-structure.
Then the following conditions are equivalent:
The Hodge structure of P is locally constant.
The representation P_s of \pi _1(S,s) factors through a finite quotient of \pi _1(S,s).
There exists a finite non-empty étale covering u \colon S' \to S such that u^*P is a constant family of Hodge structures.
1648hodge-theory-ii-4.2hodge-theory-ii-4.2.xmlThe semi-simplicity theorem4.2hodge-theory-ii-41102hodge-theory-ii-4.2.1hodge-theory-ii-4.2.1.xml4.2.1hodge-theory-ii-4.2
Let S be a topological space.
A continuous family of Hodge structures on S consists of:
A local system H_ \mathbb {Z} on S of \mathbb {Z}-modules of finite type.
For every point s \in S, a Hodge structure on the fibre (H_ \mathbb {Z} ) that varies continuously in s.
A continuous family H of Hodge structures on S is said to be of weight n if the fibres H_s (for s \in S) are all of weight n.
We similarly define a continuous family of Hodge \mathbb {Q}-structures as a local system of \mathbb {Q}-vector spaces, endowed at each point with a Hodge \mathbb {Q}-structure, varying continuously.
A polarisation of a continuous family H of Hodge \mathbb {Q}-structures of weight n is a morphism of local systems from H_ \mathbb {Q} \otimes H_ \mathbb {Q} to the constant local system \mathbb {Q} (-n)_ \mathbb {Q}, which defines at each point s \in S a polarisation of H_s.
1107hodge-theory-ii-4.2.2hodge-theory-ii-4.2.2.xml4.2.2hodge-theory-ii-4.2
Suppose that S is connected, and let \mathcal {C} be a strictly full subcategory of the category of continuous families of Hodge \mathbb {Q}-structures on S.
We will have need of the following conditions:
1103Conditionhodge-theory-ii-4.2.2.1hodge-theory-ii-4.2.2.1.xml4.2.2.1hodge-theory-ii-4.2.2\mathcal {C} is stable under direct factors, direct sums, and tensor products;
the Tate constant families \mathbb {Q} (n) (for n \in \mathbb {Z}) are in \mathcal {C}.
1104Conditionhodge-theory-ii-4.2.2.2hodge-theory-ii-4.2.2.2.xml4.2.2.2hodge-theory-ii-4.2.2
Every homogeneous (of some weight n) Hodge structure in \mathcal {C} is polarisable.
1105Conditionhodge-theory-ii-4.2.2.3hodge-theory-ii-4.2.2.3.xml4.2.2.3hodge-theory-ii-4.2.2
For every H \in \operatorname {Ob} \mathcal {C}, there exists a local system H'_ \mathbb {Z} on S of free \mathbb {Z}-modules such that H'_ \mathbb {Z} \otimes \mathbb {Q} \approx H_ \mathbb {Q}.
1106Conditionhodge-theory-ii-4.2.2.4hodge-theory-ii-4.2.2.4.xml4.2.2.4hodge-theory-ii-4.2.2
For every H \in \operatorname {Ob} \mathcal {C}, the largest constant local system H^f of H is a constant family of Hodge sub-structures of H.
1112Lemmahodge-theory-ii-4.2.3hodge-theory-ii-4.2.3.xml4.2.3hodge-theory-ii-4.2
If \mathcal {C} satisfies and , then:
\mathcal {C} is a semi-simple abelian subcategory of the abelian subcategory of continuous families of Hodge \mathbb {Q}-structures on S.
If H \in \operatorname {Ob} \mathcal {C}, then its dual H^* and the \wedge ^p H are also in \mathcal {C};
if H_1,H_2 \in \operatorname {Ob} \mathcal {C}, then \operatorname {Hom} (H_1,H_2) \in \operatorname {Ob} \mathcal {C}.
1111Proofhodge-theory-ii-4.2.3
Let H \in \operatorname {Ob} \mathcal {C}, and let H_1 be a sub-object of H, in the category of continuous families of Hodge \mathbb {Q}-structures.
We will show that H_1 is a direct factor of H in this category.
We can suppose H to be homogeneous.
If \psi is a polarisation form for H, then the orthogonal of H_1 with respect to \psi is indeed a sub-object of H that is a supplement to H_1.
This proves (i)
If H \in \operatorname {Ob} \mathcal {C} is of weight n, then a polarisation of H defines an isomorphism between H^* and H \otimes \mathbb {Q} (n).
By , we thus have that H^* \in \operatorname {Ob} \mathcal {C}.
For arbitrary H \in \operatorname {Ob} \mathcal {C}, if we decompose H into its homogeneous components, then H^*= \bigoplus _n(H^n)^*, whence again H^* \in \operatorname {Ob} \mathcal {C}.
Finally, \wedge ^p H is a direct factor of \bigotimes ^p H, and \operatorname {Hom} (H_1,H_2) \simeq H_1^* \otimes H_2.
Let f \colon X \to S be a smooth proper morphism of reduced schemes.
The sheaf \mathrm {R} ^i f_* \mathbb {Q} is then a local system, and, for s \in S, ( \mathrm {R} ^i f_* \mathbb {Q} )_s \simeq \operatorname {H} ^i(X_s, \mathbb {Q} ) is endowed with a Hodge \mathbb {Q}-structure, which varies continuously in s.
1113Definitionhodge-theory-ii-4.2.4hodge-theory-ii-4.2.4.xml4.2.4hodge-theory-ii-4.2
Let S be a smooth connected scheme.
A continuous family H of Hodge \mathbb {Q}-structures on S^ \mathrm {an} is said to be algebraic if there exists a non-empty Zariski open U of S, an integer k, and a smooth projective morphism f \colon X \to U such that H|U is a direct factor of \mathrm {R} f_* \mathbb {Q} \otimes \mathbb {Q} (k).
1125Propositionhodge-theory-ii-4.2.5hodge-theory-ii-4.2.5.xml4.2.5hodge-theory-ii-4.2
The category of algebraic continuous families of Hodge \mathbb {Q}-structures on S satisfies the conditions of .
If a continuous family H of Hodge structures on S is such that its restriction to a dense Zariski open U of S is algebraic, then H is algebraic.
If f \colon X \to S is smooth and proper, then \mathrm {R} f_* \mathbb {Q} is algebraic.
1124Proofhodge-theory-ii-4.2.5
Claim (ii) is evident.
We will prove (i).
Let \mathcal {C}_0 be the set of continuous families of Hodge structures on S that are of the form \mathrm {R} f_* \mathbb {Q} (with f smooth and projective).
Then:
By the Künneth formula
\mathrm {R} (f \times g)_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Q} \otimes \mathrm {R} g_* \mathbb {Q} ,
\mathcal {C}_0 is stable under tensor products.
Since
\mathrm {R} (f \sqcup g)_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Q} \oplus \mathrm {R} g_* \mathbb {Q} ,
\mathcal {C}_0 is stable under direct sums.
The homogeneous components of each H \in \operatorname {Ob} \mathcal {C}_0 are polarisable (cf. ).
The objects H \in \operatorname {Ob} \mathcal {C}_0 satisfy , since
\mathrm {R} f_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Z} \otimes \mathbb {Q} .
The objects H \in \operatorname {Ob} \mathcal {C}_0 satisfy , by .
Let \mathcal {C}_1 be the set of direct factors of objects of \mathcal {C}_0.
Then \mathcal {C}_1 is stable under tensor products, direct sums, and direct factors, and satisfies , , and .
Furthermore, \mathbb {Q} (-1) is in \mathcal {C}_1, in the form of \mathrm {R} ^2 f_* \mathbb {Q} for f \colon \mathbb {P} _S^1 \to S the projective bundle.
The category \mathcal {C}_2 consisting of the H \otimes \mathbb {Q} (k) (for k \in \mathbb {N}) thus satisfy all the conditions of .
To thus deduce (i), it suffices to note that, if H is a continuous family of Hodge structures, and U a dense Zariski open of S, then a sub-object of, a polarisation on, or an integer lattice of H|U uniquely extend to H.
We now prove (iii).
If f \colon X \to S is smooth and projective, then there exists a dense Zariski open subset U of S along with a commutative diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X' \ar [rr,"p"] \ar [dr,swap,"f'"]
&& X|U \ar [dl,"f|U"]
\\ & U
\end {tikzcd}
with f' smooth and projective, and p birational and surjective (by applying Chow's lemma and the resolution of singularities to the generic fibre of f).
The restriction map
p^* \colon ( \mathrm {R} f_* \mathbb {Q} )|U \to \mathrm {R} f'_* \mathbb {Q}
is thus a direct injection of continuous families of Hodge structures, with left inverse given by the Gysin morphism p_!, whence the algebraicity of \mathrm {R} f_* \mathbb {Q}.
1126Theoremhodge-theory-ii-4.2.6hodge-theory-ii-4.2.6.xml4.2.6hodge-theory-ii-4.2
Let S be a connected topological space that is locally connected and locally simply connected, endowed with a basepoint s.
Let \mathcal {C} be a category of continuous Hodge structures on S that satisfy the conditions of .
Then, if H \in \operatorname {Ob} \mathcal {C}, the representation of \pi _1(S,s) on the fibre (H_ \mathbb {Q} )_s is semisimple.
1127Footnotehodge-theory-ii-4.2.6-footnote-1hodge-theory-ii-4.2.6-footnote-1.xml1hodge-theory-ii-4.2
Let S be a smooth connected scheme.
A family of Hodge structures on S is a continuous family H of Hodge structures on S that satisfies the following conditions:
The Hodge filtration on (H_ \mathbb {C} )_s varies holomorphically with s, i.e. it corresponds to a filtration F of H_ \mathcal {O} =H_ \mathbb {Z} \otimes \mathcal {O}.
The covariant derivative \nabla satisfies \nabla F^p \subset \Omega _S^1 \otimes F^{p-1}.
Let \mathcal {C} be the category of such continuous families of Hodge \mathbb {Q}-structures on S that underlie a family of Hodge structures and whose homogeneous direct factors are polarisable.
It is clear that \mathcal {C} satisfies , , and .
We can deduce from results of W. Schmid (August 1970, unpublished) and P.A. Griffiths [G1970] that \mathcal {C} satisfies .
This theorem, of which is a corollary, allows us to apply and its corollaries to the objects of \mathcal {C}.
Let H \in \operatorname {Ob} \mathcal {C}.
The local system H_ \mathbb {Q} defines a complex local system H_ \mathbb {C} =H_ \mathbb {Q} \otimes \mathbb {C}.
For every complex local system V on S, we denote by V^c the complex vector bundle that it defines, identified with the sheaf of its continuous sections.
By definition, the group S (from ) acts on H_ \mathbb {C} ^c.
A sub-bundle of H_ \mathbb {C} ^c is said to be horizontal, or locally constant, if it is defined by a local subsystem of H_ \mathbb {C}.
1129Lemmahodge-theory-ii-4.2.7hodge-theory-ii-4.2.7.xml4.2.7hodge-theory-ii-4.2
Let V be a rank-1 local subsystem of H_ \mathbb {C}.
Suppose that the n-th tensor power V^{ \otimes n} (for some n \geqslant1) of V is a trivial local system, i.e. that \pi _1(S,s) acts on V_s via a finite (necessarily cyclic) group.
Then, for all t \in S, tV^c is again locally constant.
1128Proofhodge-theory-ii-4.2.7
In order for tV^c to be locally constant, it suffices for (tV^c)^{ \otimes n}={tV^{c}}^{ \otimes n} \subset \bigotimes ^n H_ \mathbb {C} to be locally constant.
But V^{ \otimes n} is generated by a horizontal global section v, and, by hypothesis, tv is again horizontal .
We proceed by induction on \dim (H_ \mathbb {Q} )_s.
We can suppose H to be homogeneous and non-zero.
Let d be the minimal dimension of the non-zero complex local subsystems of H_ \mathbb {C}.
The sum W of all the local subsystems of H_ \mathbb {C} of dimension d (which are automatically simple) is "defined over \mathbb {Q}", i.e. is of the form W_ \mathbb {Q} \otimes \mathbb {C} for W_ \mathbb {Q} a local subsystem of H_ \mathbb {Q}.
By construction, W_s is a semisimple complex \pi _1(S,s)-module, so (W_ \mathbb {Q} )_s is a semisimple \pi _1(S,s)-module over \mathbb {Q}.
Let
1649hodge-theory-ii-4.3hodge-theory-ii-4.3.xmlSupplement to [D1968]4.3hodge-theory-ii-41650hodge-theory-ii-4.4hodge-theory-ii-4.4.xmlHomomorphisms from abelian schemes4.4hodge-theory-ii-41692Bibliographyhodge-theory-bibliographyhodge-theory-bibliography.xmlComplete list of referencesClick anywhere on an entry to see the corresponding BibTeX.
16891653ReferenceK1971K1971.xmlNilpotent connections and the monodromy theorem. Applications of a result of Turritin1971N. KatzPubl. Math. IHÉS 39 pp. 175–232@article{10,
author = {Katz, N.},
title = {{Nilpotent connections and the monodromy theorem. Applications of a result of Turritin}},
journal = {Publ. Math. IH\'{E}S},
volume = {39},
year = {1971},
pages = {175--232},
}1655ReferenceD1970D1970.xmlEquations différentielles à points singuliers réguliers1970P. DeligneSpringer@book{3,
author = {Deligne, P.},
title = {Equations diff\'{e}rentielles \`{a} points singuliers r\'{e}guliers},
publisher = {Springer},
year = {1970},
}1657ReferenceG1970G1970.xmlPeriods of integrals on algebraic manifolds, III1970P.A. GriffithsPubl. Math. IHÉS 38 pp. 125–180@article{5,
author = {Griffiths, P.A.},
title = {Periods of integrals on algebraic manifolds, {III}},
journal = {Publ. Math. IH\'{E}S},
volume = {38},
year = {1970},
pages = {125--180},
}1659ReferenceD1969D1969.xmlMotifs des variétés algébriques1969M. DemazureSém. Bourbaki (1969–70) no. 365@article{D1969,
author = {Demazure, M},
title = {Motifs des vari\'{e}t\'{e}s alg\'{e}briques},
journal = {S\'{e}m. Bourbaki},
year = {1969--70},
pages = {Talk no.~365},
}1661ReferenceG1969G1969.xmlOn the periods of certain rational integrals, I1969P.A. GriffithsAnn. of Math. 90 pp. 460–495@article{4,
author = {Griffiths, P.A.},
title = {On the periods of certain rational integrals, {I}},
journal = {Ann. of Math.},
volume = {90},
year = {1969},
pages = {460--495},
}1663ReferenceST1968ST1968.xmlGood reduction of abelian varieties1968J.-P. SerreJ. TateAnn. of Math 88 pp. 492–517@article{ST1968,
author = {Serre, J-P and Tate, J},
title = {Good reduction of abelian varieties},
journal = {Ann. of Math.},
volume = {88},
year = {1968},
pages = {392--517},
}1666ReferenceG1968G1968.xmlLe groupe de Brauer, III: Exemples et compléments1968A. GrothendieckDix exposés sur la cohomologie des schémas North-Holland Publ. Co.@incollection{6,
author = {Grothendieck, A.},
title = {{Le groupe de Brauer, III: Exemples et compl\'{e}ments}},
booktitle = {Dix expos\'{e}s sur la cohomologie des sch\'{e}mas},
publisher = {North-Holland Publ. Co.},
year = {1968},
}1668ReferenceD1968D1968.xmlThéorème de Lefschetz et critères de dégénérescence de suites spectrales1968P. DelignePubl. Math. IHÉS 35 pp.107–126@article{2,
author = {Deligne, P.},
title = {Th\'{e}or\`{e}me de {Lefschetz} et crit\`{e}res de d\'{e}g\'{e}n\'{e}rescence de suites spectrales},
journal = {Publ. Math. IH\'{E}S},
volume = {35},
year = {1968},
pages = {107--126},
}1670ReferenceG1966G1966.xmlUn théorème sur les homomorphismes de schémas abéliens1966A. GrothendieckInv. Math. 2 pp. 59–78@article{7,
author = {Grothendieck, A.},
title = {Un th\'{e}or\`{e}me sur les homomorphismes de sch\'{e}mas ab\'{e}liens},
journal = {Inv. Math.},
volume = {2},
year = {1966},
pages = {59--78},
}1672ReferenceH1964H1964.xmlResolution of singularities of an algebraic variety over a field of characteristic zero1964H. HironakaAnn. of Math. 79 pp. 109–326@article{8,
author = {Hironaka, H.},
title = {Resolution of singularities of an algebraic variety over a field of characteristic zero},
journal = {Ann. of Math.},
volume = {79},
year = {1964},
pages = {109--326},
}1674ReferenceN1962N1962.xmlImbedding of an abstract variety in a complete variety1962M. NagataJ. Math. Kyoto 2 pp. 1–10@article{11,
author = {Nagata, M.},
title = {Imbedding of an abstract variety in a complete variety},
journal = {J. Math. Kyoto},
volume = {2},
year = {1962},
pages = {1-10},
}1676ReferenceS1960S1960.xmlAnalogues kählériens de certaines conjectures de Weil1960J.-P. SerreAnn. of Math 71 pp. 392–294@article{3,
author = {Serre, J-P},
title = {Analogues k\"{a}hl\'{e}riens de certaines conjectures de Weil},
journal = {Ann. of Math.},
volume = {71},
year = {1960},
pages = {392--394},
}1678ReferenceW1958W1958.xmlIntroduction à L'Étude des Variétés Kählériennes1958A. WeilHermann Act. Sci. et Ind. 1267@book{W1958,
author = {Weil, A},
title = {Introduction \`{a} L'\'{E}tude des Vari\'{e}t\'{e}s K\"{a}hl\'{e}riennes},
publisher = {Hermann},
year = {1958},
series = {Act. Sci. et Ind.},
volume = {1267},
}1680ReferenceG1958G1958.xmlTopologie algébrique et théorie des faisceaux1958R. GodementHermann@book{15,
author = {Godement, R.},
title = {Topologie alg\'{e}brique et th\'{e}orie des faisceaux},
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}1682ReferenceAH1955AH1955.xmlIntegrals of the second kind on an algebraic variety1955M.F. AtiyahW.V.D. HodgeAnn. of Math. 62 pp. 56–91@article{1,
author = {Atiyah, M.F. and Hodge, W.V.D.},
title = {Integrals of the second kind on an algebraic variety},
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}1685ReferenceH1952H1952.xmlThe theory and applications of harmonic integrals1952W.V.D. HodgeCambridge Univ. Press@book{9,
author = {Hodge, W.V.D.},
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author = {Weil, A},
title = {Number of solutions of equations in finite fields},
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1693hodge-theory-ii-1.1hodge-theory-ii-1.1.xmlHodge Theory II › Filtrations › Filtered objects1.1hodge-theory-ii-1579hodge-theory-ii-1.1.1hodge-theory-ii-1.1.1.xml1.1.1hodge-theory-ii-1.1
Let \mathscr {A} be an abelian category.
We will be considering \mathbb {Z}-filtrations, finite, in general, on the objects of \mathscr {A}:
580Definitionhodge-theory-ii-1.1.2hodge-theory-ii-1.1.2.xml1.1.2hodge-theory-ii-1.1
A decreasing (resp. increasing) filtration F of an object A of \mathscr {A} is a family (F^n(A))_{n \in \mathbb {Z} } (resp. (F_n(A))_{n \in \mathbb {Z} }) of sub-objects of A satisfying
\forall n,m \quad n \leqslant m \implies F^m(A) \subset F^n(A)
(resp. n \leqslant m \implies F_n(A) \subset F_m(A)).
A filtered object is an object endowed with a filtration.
When there is no chance of confusion, we often denote by the same letter filtrations on different objects of \mathscr {A}.
If F is a decreasing (resp. increasing) filtration on A, then we set F^ \infty (A)=0 and F^{- \infty }(A)=A (resp. F_{- \infty }(A)=0 and F_ \infty (A)=A).
The shifted filtrations of a decreasing filtration W are defined by
W[n]^p(A) = W^{n+p}(A)
for n \in \mathbb {Z}.
581hodge-theory-ii-1.1.3hodge-theory-ii-1.1.3.xml1.1.3hodge-theory-ii-1.1
If R is a decreasing (resp. increasing) filtration of A, then the F_n(A)=F^{-n}(A) (resp the F^n(A)=F_{-n}(A)) form an increasing (resp. decreasing) filtration of A.
This allows us in principal to consider only decreasing filtrations;
unless otherwise explicitly mentioned, when we say "filtration" we always mean "decreasing filtration".
582hodge-theory-ii-1.1.4hodge-theory-ii-1.1.4.xml1.1.4hodge-theory-ii-1.1
A filtration F of A is said to be finite if there exist n and m such that F^n(A)=A and F^m(A)=0.
583hodge-theory-ii-1.1.5hodge-theory-ii-1.1.5.xml1.1.5hodge-theory-ii-1.1
A morphism from a filtered object (A,F) to a filtered object (B,F) is a morphism f from A to B that satisfies f(F^n(A)) \subset F^n(B) for all n \in \mathbb {Z}.
Filtered objects (resp. finite filtered objects) of \mathscr {A} form an additive category in which inductive limits and finite projective limits exist (and thus kernels, cokernels, images, and coimages of a morphism).
A morphism f \colon (A,F) \to (B,F) is said to be strict, or strictly compatible with the filtrations, if the canonical arrows from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism of filtered objects (cf. ).
584hodge-theory-ii-1.1.6hodge-theory-ii-1.1.6.xml1.1.6hodge-theory-ii-1.1
Let (-)^ \circ be the contravariant identity functor from \mathscr {A} to the dual category \mathscr {A}^ \circ.
If (A,F) is a filtered object of \mathscr {A}, then the (A/F^n(A))^ \circ can be identified with sub-objects of A^ \circ.
The dual filtration on A^ \circ is defined by
F^n(A^ \circ ) = (A/F^{1-n})^ \circ .
The double dual of (A,F) can be identified with (A,F).
This construction identifies the dual of the category of filtered objects of \mathscr {A} with the category of filtered objects of \mathscr {A}^ \circ.
586hodge-theory-ii-1.1.7hodge-theory-ii-1.1.7.xml1.1.7hodge-theory-ii-1.1
If (A,F) is a filtered object of \mathscr {A}, then its associated graded is the object of \mathscr {A}^ \mathbb {Z} defined by
\operatorname {Gr} ^n(A) = F^n(A)/F^{n+1}(A).
The convention is justified by the simple formula
\operatorname {Gr} ^n(A^ \circ ) = \operatorname {Gr} ^{-n}(A)^ \circ
which follows from the self-dual diagram
585Equationhodge-theory-ii-1.1.7.1hodge-theory-ii-1.1.7.1.xml1.1.7.1hodge-theory-ii-1.1.7
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/F^n(A)
&& A/ F^{n+1}(A)
\ar [ll]
&& 0
\\ & A
\ar [ul] \ar [ur]
&& \operatorname {Gr} ^n(A)
\ar [ul] \ar [ur]
&& (1.1.7.1)
\\ F^{n+1}(A)
\ar [ur] \ar [rr]
&& F^n(A)
\ar [ul] \ar [ur]
&& 0
\ar [ul]
\end {tikzcd}
587hodge-theory-ii-1.1.8hodge-theory-ii-1.1.8.xml1.1.8hodge-theory-ii-1.1
Let (A,F) be a filtered object, and j \colon X \hookrightarrow A a sub-object of A.
The filtration induced by F (or simply the induced filtration) on X is the unique filtration on X such that j is strictly compatible with the filtrations;
we have
F^n(X) = j^{-1}(F^n(A)) = X \cap F^n(A).
Dually, the quotient filtration on A/X (the unique filtration such that p \colon A \to A/X is strictly compatible with the filtrations) is given by
F^n(A/X) = p(F^n(A)) \cong (X+F^n(A))/X \cong F^n(A)/(X \cap F^n(A)). 588Lemmahodge-theory-ii-1.1.9hodge-theory-ii-1.1.9.xml1.1.9hodge-theory-ii-1.1
If X and Y are sub-objects of A, with X \subset Y, then on Y/X \xrightarrow { \sim } \operatorname {Ker} (A/X \to A/Y) the quotient filtration of Y agrees with that induced by that of A/X.
In the diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/Y
&& A/X
\ar [ll]
\\ & A
\ar [ul] \ar [ur]
&& Y/X
\ar [ul]
\\ X
\ar [rr] \ar [ur]
&& Y
\ar [ul] \ar [uu] \ar [ur]
\end {tikzcd}
the arrows are strict.
589hodge-theory-ii-1.1.10hodge-theory-ii-1.1.10.xml1.1.10hodge-theory-ii-1.1
We call the filtration on Y/X the filtration induced by that of A (or simply the induced filtration).
By , its definition is self-dual.
In particular, if \Sigma \colon A \xrightarrow {f}B \xrightarrow {G}C is a **!!TO-DO: o-suite?!!** sequence, and if B is filtered, then \operatorname {H} ( \Sigma )= \operatorname {Ker} (g)/ \operatorname {Im} (f)= \operatorname {Ker} ( \operatorname {Coker} (f) \to \operatorname {Coim} (g)) is endowed with a canonical induced filtration.
The reader can show that:
593Propositionhodge-theory-ii-1.1.11hodge-theory-ii-1.1.11.xml1.1.11hodge-theory-ii-1.1
Let f \colon (A,F) \to (B,F) be a morphism of filtered objects with finite filtrations.
For f to be strict, it is necessary and sufficient that the sequence
0 \to \operatorname {Gr} ( \operatorname {Ker} (f)) \to \operatorname {Gr} (A) \to \operatorname {Gr} (B) \to \operatorname {Gr} ( \operatorname {Coker} (f)) \to 0
be exact.
Let \sigma \colon (A,F) \to (B,F) \to (C,F) be a **!!TO-DO: o-suite?!!** sequence of strict morphisms.
We then have
\operatorname {H} ( \operatorname {Gr} ( \Sigma )) \cong \operatorname {Gr} ( \operatorname {H} ( \Sigma ))
canonically.
In particular, if \Sigma is exact in \mathscr {A}, then \operatorname {Gr} ( \Sigma ) is exact in \mathscr {A}^ \mathbb {Z}.
In a category of modules, to say that a morphism f \colon (A,F) \to (B,F) is strict implies that every b \in B of filtration \geqslant n (i.e. b \in F^n(B)) that is in the image of A is already in the image of F^n(A):
f(F^n(A)) = f(A) \cap F^n(B). 594hodge-theory-ii-1.1.12hodge-theory-ii-1.1.12.xml1.1.12hodge-theory-ii-1.1
If \otimes \colon \mathscr {A}_1 \times \ldots \times \mathscr {A}_n \to \mathscr {B} is a multiadditive right-exact functor, and if A_i is an object of finite filtration of \mathscr {A}_i (for 1 \leqslant i \leqslant n), then we define a filtration on \bigotimes _{i=1}^n A_i by
F^k( \bigotimes _{i=1}^n A_i) = \sum _{ \sum k_i=k} \operatorname {Im} ( \bigotimes _{i=1}^n F^{k_i}(A_i) \to \bigotimes _{i=1}^n A_i)
(a sum of sub-objects).
Dually, if H is left-exact, then we set
F^k(H(A_i)) = \bigcap _{ \sum k_i=k} \operatorname {Ker} (H(A_i) \to H(A_i/F^{k_i}(A_i))).
If H is exact, then the two definitions are equivalent.
We extend these definitions to contravariant functors in certain variables by .
In particular, for the left-exact functor \operatorname {Hom}, we set
F^k( \operatorname {Hom} (A,B)) = \{ f \colon A \to B \mid f(F^n(A)) \subset F^{n+k}(B) \, \, \forall n \} .
We thus have
\operatorname {Hom} ((A,F),(B,F)) = F^0( \operatorname {Hom} (A,B)).
Under the above hypotheses, we have obvious morphisms
\begin {aligned} \bigotimes _{i=1}^n \operatorname {Gr} (A_i) & \to \operatorname {Gr} ( \bigotimes _{i=1}^n A_i) \\ \operatorname {Gr} H(A_i) & \to H( \operatorname {Gr} (A_i)). \end {aligned}
If H is exact, then these are isomorphisms and inverse to one another.
These constructions are compatible with composition of functors, in a sense whose details we leave to the reader.
1694hodge-theory-ii-1hodge-theory-ii-1.xmlHodge Theory II › Filtrations1hodge-theory-ii1634hodge-theory-ii-1.1hodge-theory-ii-1.1.xmlFiltered objects1.1hodge-theory-ii-1579hodge-theory-ii-1.1.1hodge-theory-ii-1.1.1.xml1.1.1hodge-theory-ii-1.1
Let \mathscr {A} be an abelian category.
We will be considering \mathbb {Z}-filtrations, finite, in general, on the objects of \mathscr {A}:
580Definitionhodge-theory-ii-1.1.2hodge-theory-ii-1.1.2.xml1.1.2hodge-theory-ii-1.1
A decreasing (resp. increasing) filtration F of an object A of \mathscr {A} is a family (F^n(A))_{n \in \mathbb {Z} } (resp. (F_n(A))_{n \in \mathbb {Z} }) of sub-objects of A satisfying
\forall n,m \quad n \leqslant m \implies F^m(A) \subset F^n(A)
(resp. n \leqslant m \implies F_n(A) \subset F_m(A)).
A filtered object is an object endowed with a filtration.
When there is no chance of confusion, we often denote by the same letter filtrations on different objects of \mathscr {A}.
If F is a decreasing (resp. increasing) filtration on A, then we set F^ \infty (A)=0 and F^{- \infty }(A)=A (resp. F_{- \infty }(A)=0 and F_ \infty (A)=A).
The shifted filtrations of a decreasing filtration W are defined by
W[n]^p(A) = W^{n+p}(A)
for n \in \mathbb {Z}.
581hodge-theory-ii-1.1.3hodge-theory-ii-1.1.3.xml1.1.3hodge-theory-ii-1.1
If R is a decreasing (resp. increasing) filtration of A, then the F_n(A)=F^{-n}(A) (resp the F^n(A)=F_{-n}(A)) form an increasing (resp. decreasing) filtration of A.
This allows us in principal to consider only decreasing filtrations;
unless otherwise explicitly mentioned, when we say "filtration" we always mean "decreasing filtration".
582hodge-theory-ii-1.1.4hodge-theory-ii-1.1.4.xml1.1.4hodge-theory-ii-1.1
A filtration F of A is said to be finite if there exist n and m such that F^n(A)=A and F^m(A)=0.
583hodge-theory-ii-1.1.5hodge-theory-ii-1.1.5.xml1.1.5hodge-theory-ii-1.1
A morphism from a filtered object (A,F) to a filtered object (B,F) is a morphism f from A to B that satisfies f(F^n(A)) \subset F^n(B) for all n \in \mathbb {Z}.
Filtered objects (resp. finite filtered objects) of \mathscr {A} form an additive category in which inductive limits and finite projective limits exist (and thus kernels, cokernels, images, and coimages of a morphism).
A morphism f \colon (A,F) \to (B,F) is said to be strict, or strictly compatible with the filtrations, if the canonical arrows from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism of filtered objects (cf. ).
584hodge-theory-ii-1.1.6hodge-theory-ii-1.1.6.xml1.1.6hodge-theory-ii-1.1
Let (-)^ \circ be the contravariant identity functor from \mathscr {A} to the dual category \mathscr {A}^ \circ.
If (A,F) is a filtered object of \mathscr {A}, then the (A/F^n(A))^ \circ can be identified with sub-objects of A^ \circ.
The dual filtration on A^ \circ is defined by
F^n(A^ \circ ) = (A/F^{1-n})^ \circ .
The double dual of (A,F) can be identified with (A,F).
This construction identifies the dual of the category of filtered objects of \mathscr {A} with the category of filtered objects of \mathscr {A}^ \circ.
586hodge-theory-ii-1.1.7hodge-theory-ii-1.1.7.xml1.1.7hodge-theory-ii-1.1
If (A,F) is a filtered object of \mathscr {A}, then its associated graded is the object of \mathscr {A}^ \mathbb {Z} defined by
\operatorname {Gr} ^n(A) = F^n(A)/F^{n+1}(A).
The convention is justified by the simple formula
\operatorname {Gr} ^n(A^ \circ ) = \operatorname {Gr} ^{-n}(A)^ \circ
which follows from the self-dual diagram
585Equationhodge-theory-ii-1.1.7.1hodge-theory-ii-1.1.7.1.xml1.1.7.1hodge-theory-ii-1.1.7
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/F^n(A)
&& A/ F^{n+1}(A)
\ar [ll]
&& 0
\\ & A
\ar [ul] \ar [ur]
&& \operatorname {Gr} ^n(A)
\ar [ul] \ar [ur]
&& (1.1.7.1)
\\ F^{n+1}(A)
\ar [ur] \ar [rr]
&& F^n(A)
\ar [ul] \ar [ur]
&& 0
\ar [ul]
\end {tikzcd}
587hodge-theory-ii-1.1.8hodge-theory-ii-1.1.8.xml1.1.8hodge-theory-ii-1.1
Let (A,F) be a filtered object, and j \colon X \hookrightarrow A a sub-object of A.
The filtration induced by F (or simply the induced filtration) on X is the unique filtration on X such that j is strictly compatible with the filtrations;
we have
F^n(X) = j^{-1}(F^n(A)) = X \cap F^n(A).
Dually, the quotient filtration on A/X (the unique filtration such that p \colon A \to A/X is strictly compatible with the filtrations) is given by
F^n(A/X) = p(F^n(A)) \cong (X+F^n(A))/X \cong F^n(A)/(X \cap F^n(A)). 588Lemmahodge-theory-ii-1.1.9hodge-theory-ii-1.1.9.xml1.1.9hodge-theory-ii-1.1
If X and Y are sub-objects of A, with X \subset Y, then on Y/X \xrightarrow { \sim } \operatorname {Ker} (A/X \to A/Y) the quotient filtration of Y agrees with that induced by that of A/X.
In the diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/Y
&& A/X
\ar [ll]
\\ & A
\ar [ul] \ar [ur]
&& Y/X
\ar [ul]
\\ X
\ar [rr] \ar [ur]
&& Y
\ar [ul] \ar [uu] \ar [ur]
\end {tikzcd}
the arrows are strict.
589hodge-theory-ii-1.1.10hodge-theory-ii-1.1.10.xml1.1.10hodge-theory-ii-1.1
We call the filtration on Y/X the filtration induced by that of A (or simply the induced filtration).
By , its definition is self-dual.
In particular, if \Sigma \colon A \xrightarrow {f}B \xrightarrow {G}C is a **!!TO-DO: o-suite?!!** sequence, and if B is filtered, then \operatorname {H} ( \Sigma )= \operatorname {Ker} (g)/ \operatorname {Im} (f)= \operatorname {Ker} ( \operatorname {Coker} (f) \to \operatorname {Coim} (g)) is endowed with a canonical induced filtration.
The reader can show that:
593Propositionhodge-theory-ii-1.1.11hodge-theory-ii-1.1.11.xml1.1.11hodge-theory-ii-1.1
Let f \colon (A,F) \to (B,F) be a morphism of filtered objects with finite filtrations.
For f to be strict, it is necessary and sufficient that the sequence
0 \to \operatorname {Gr} ( \operatorname {Ker} (f)) \to \operatorname {Gr} (A) \to \operatorname {Gr} (B) \to \operatorname {Gr} ( \operatorname {Coker} (f)) \to 0
be exact.
Let \sigma \colon (A,F) \to (B,F) \to (C,F) be a **!!TO-DO: o-suite?!!** sequence of strict morphisms.
We then have
\operatorname {H} ( \operatorname {Gr} ( \Sigma )) \cong \operatorname {Gr} ( \operatorname {H} ( \Sigma ))
canonically.
In particular, if \Sigma is exact in \mathscr {A}, then \operatorname {Gr} ( \Sigma ) is exact in \mathscr {A}^ \mathbb {Z}.
In a category of modules, to say that a morphism f \colon (A,F) \to (B,F) is strict implies that every b \in B of filtration \geqslant n (i.e. b \in F^n(B)) that is in the image of A is already in the image of F^n(A):
f(F^n(A)) = f(A) \cap F^n(B). 594hodge-theory-ii-1.1.12hodge-theory-ii-1.1.12.xml1.1.12hodge-theory-ii-1.1
If \otimes \colon \mathscr {A}_1 \times \ldots \times \mathscr {A}_n \to \mathscr {B} is a multiadditive right-exact functor, and if A_i is an object of finite filtration of \mathscr {A}_i (for 1 \leqslant i \leqslant n), then we define a filtration on \bigotimes _{i=1}^n A_i by
F^k( \bigotimes _{i=1}^n A_i) = \sum _{ \sum k_i=k} \operatorname {Im} ( \bigotimes _{i=1}^n F^{k_i}(A_i) \to \bigotimes _{i=1}^n A_i)
(a sum of sub-objects).
Dually, if H is left-exact, then we set
F^k(H(A_i)) = \bigcap _{ \sum k_i=k} \operatorname {Ker} (H(A_i) \to H(A_i/F^{k_i}(A_i))).
If H is exact, then the two definitions are equivalent.
We extend these definitions to contravariant functors in certain variables by .
In particular, for the left-exact functor \operatorname {Hom}, we set
F^k( \operatorname {Hom} (A,B)) = \{ f \colon A \to B \mid f(F^n(A)) \subset F^{n+k}(B) \, \, \forall n \} .
We thus have
\operatorname {Hom} ((A,F),(B,F)) = F^0( \operatorname {Hom} (A,B)).
Under the above hypotheses, we have obvious morphisms
\begin {aligned} \bigotimes _{i=1}^n \operatorname {Gr} (A_i) & \to \operatorname {Gr} ( \bigotimes _{i=1}^n A_i) \\ \operatorname {Gr} H(A_i) & \to H( \operatorname {Gr} (A_i)). \end {aligned}
If H is exact, then these are isomorphisms and inverse to one another.
These constructions are compatible with composition of functors, in a sense whose details we leave to the reader.
1635hodge-theory-ii-1.2hodge-theory-ii-1.2.xmlOpposite filtrations1.2hodge-theory-ii-1657hodge-theory-ii-1.2.1hodge-theory-ii-1.2.1.xml1.2.1hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with filtrations F and G.
By definition, \operatorname {Gr} _F^n(A) is a quotient of a sub-object of A and, as such, is endowed with a filtration induced by G .
Passing to the associated graded defines a bigraded object ( \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A))_{n,m \in \mathbb {Z} }.
By a lemma of Zassenhaus, \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) and \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) are canonically isomorphic: if we define the induced filtrations as quotient filtrations of the induced filtrations on a sub-object, then we have
\begin {aligned} \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) \cong (F^m(A) \cap G^n(A)) &/ ((F^{m+1}(A) \cap G^n(A)) + (F^m(A) \cap G^{n+1}(A))) \\ &= \\ \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) \cong (G^n(A) \cap F^m(A)) &/ ((G^{n+1}(A) \cap F^m(A)) + (G^n(A) \cap F^{m+1}(A))) \end {aligned} 658hodge-theory-ii-1.2.2hodge-theory-ii-1.2.2.xml1.2.2hodge-theory-ii-1.2
Let H be a third filtration of A.
It induces a filtration on \operatorname {Gr} _F(A), and thus on \operatorname {Gr} _G \operatorname {Gr} _F(A).
It also induces a filtration on \operatorname {Gr} _F \operatorname {Gr} _G(A).
We note that these filtrations do not in general correspond to one another under the isomorphism .
In the expression \operatorname {Gr} _H \operatorname {Gr} _G \operatorname {Gr} _F(A), G and H thus play a symmetric role, but not F and G.
659Definitionhodge-theory-ii-1.2.3hodge-theory-ii-1.2.3.xml1.2.3hodge-theory-ii-1.2
Two finite filtrations F and \bar {F} on A are said to be n-opposite if \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q \neq n.
666hodge-theory-ii-1.2.4hodge-theory-ii-1.2.4.xml1.2.4hodge-theory-ii-1.2
If A^{p,q} is a bigraded object of \mathscr {A} such that
A^{p,q}=0 except for a finite number of pairs (p,q), and
A^{p,q}=0 for p+q \neq n
then we define two n-opposite filtrations of A= \sum _{p,q}A^{p,q} by setting
663Equationhodge-theory-ii-1.2.4.1hodge-theory-ii-1.2.4.1.xml1.2.4.1hodge-theory-ii-1.2.4 F^p(A) = \sum _{p' \geqslant p} A^{p',q'} \tag{1.2.4.1}
664Equationhodge-theory-ii-1.2.4.2hodge-theory-ii-1.2.4.2.xml1.2.4.2hodge-theory-ii-1.2.4 \bar {F}^q(A) = \sum _{q' \geqslant q} A^{p',q'}. \tag{1.2.4.2}
We have
665Equationhodge-theory-ii-1.2.4.3hodge-theory-ii-1.2.4.3.xml1.2.4.3hodge-theory-ii-1.2.4 \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A) = A^{p,q}. \tag{1.2.4.3}
Conversely:
667Propositionhodge-theory-ii-1.2.5hodge-theory-ii-1.2.5.xml1.2.5hodge-theory-ii-1.2
Let F and \bar {F} be finite filtrations on A.
For F and \bar {F} to be n-opposite, it is necessary and sufficient that, for all p,q,
[p+q=n+1] \implies [F^p(A) \oplus \bar {F}^q(A) \xrightarrow { \sim } A].
If F and \bar {F} are n-opposite, and if we set
\begin {cases} A^{p,q} = 0 & \text {for }p+q \neq n \\ A^{p,q} = F^p(A) \cap \bar {F}^q(A) & \text {for }p+q=n \end {cases}
then A is the direct sum of the A^{p,q}, and F and \bar {F} come from the bigrading A^{p,q} of A by the procedure of .
608Proofhodge-theory-ii-1.2.5
The condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n implies that F^p \cap \bar {F}^q=(F^{p+1} \cap \bar {F}^q)+(F^p \cap \bar {F}^{q+1}) for p+q>n.
By hypothesis, F^p \cap \bar {F}^q is zero for large enough p+q;
by decreasing induction, we thus deduce that the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n is equivalent to the condition F^p(A) \cap \bar {F}^q(A)=0 for p+q>n.
Dually (, , ), the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q<n is equivalent to the condition A=F^p(A)+ \bar {F}^q(A) for (1-p)+(1-q)>-n, i.e. for p+q \leqslant n+1, and the claim then follows.
If F and \bar {F} are n-opposite, then we can prove by decreasing induction on p that
604Equationhodge-theory-ii-1.2.5.1hodge-theory-ii-1.2.5.1.xml1.2.5.1hodge-theory-ii-1.2.5 \bigoplus _{p' \geqslant p} A^{p',q'} \xrightarrow { \sim } F^p(A). \tag{1.2.5.1}
For F^p(A)=0, the claim is evident.
The decomposition A=F^{p+1}(A) \oplus \bar {F}^{n-p}(A) induces on F^p(A) \supset F^{p+1}(A) a decomposition
F^p(A) = F^{p+1}(A) \oplus (F^p(A) \cap \bar {F}^{n-p}(A))
and we conclude by induction.
For p small enough, we have F^p(A)=A.
By , the A^{p,q} thus form a bigrading of A, and F satisfies .
The fact that \bar {F} satisfies then follows by symmetry.
668hodge-theory-ii-1.2.6hodge-theory-ii-1.2.6.xml1.2.6hodge-theory-ii-1.2
The constructions and establish equivalences of categories that are quasi-inverse to one another between objects of \mathscr {A} endowed with two finite n-opposite filtrations and bigraded objects of \mathscr {A} of the type considered in .
669Definitionhodge-theory-ii-1.2.7hodge-theory-ii-1.2.7.xml1.2.7hodge-theory-ii-1.2
Three finite filtrations W, F, and \bar {F} on A are said to be opposite if
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A) = 0
for p+q+n \neq0.
This condition is symmetric in F and \bar {F}.
It implies that F and \bar {F} induce on W^n(A)/W^{n+1}(A) two (-n)-opposite filtrations.
We set
A^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _F^{-p-q}(A)
whence decompositions ,
W^n(A)/W^{n+1}(A) = \bigoplus _{p+q=-n} A^{p,q} \tag{1.2.7.1}
which makes \operatorname {Gr} _W(A) into a bigraded object.
675Lemmahodge-theory-ii-1.2.8hodge-theory-ii-1.2.8.xml1.2.8hodge-theory-ii-1.2
Let W, F, and \bar {F} be three finite opposite filtrations, and \sigma a sequence (p_i,q_i)_{i \geqslant0 } pairs of integers satisfying
p_i \leqslant p_j and q_i \leqslant q_j for i \geqslant j, and
p_i+q_i=p_0+q_0-i+1 for i>0.
Set p=p_0, q=q_0, n=-p-q, and
A_ \sigma = \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap F^{p_i}(A)) \right ) \cap \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \right ).
Then the projection from W^n(A) to \operatorname {Gr} _W^n(A) induces an isomorphism
A_ \sigma \xrightarrow { \sim } A^{p,q} \subset \operatorname {Gr} _W^n(A).
674Proofhodge-theory-ii-1.2.8
We will prove by induction on k the following claim:
The projection from W^n(A)/W^{n+k} to \operatorname {Gr} _W^n(A) induces an isomorphism from
\begin {aligned} \Bigg [ & \left ( \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) + W^{n+k}(A) \right ) \\ \cap & \left ( \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) + W^{n+k}(A) \right ) \Bigg ] / W^{n+k}(A) \end {aligned} \tag{$*_k$}
to A^{p,q} \subset \operatorname {Gr} _W^n(A).
For k=1, this is exactly the definition of A^{p,q}.
By (i) of we have
673Equationhodge-theory-ii-1.2.8.1hodge-theory-ii-1.2.8.1.xml1.2.8.1hodge-theory-ii-1.2.8 F^{p_k}( \operatorname {Gr} _W^{n+k}(A)) \oplus \bar {F}^{q_k}( \operatorname {Gr} _W^{n+k}(A)) \xrightarrow { \sim } \operatorname {Gr} _W^{n+k}(A). \tag{1.2.8.1}
Set
\begin {aligned} B &= \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) \\ C &= \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \\ B' &= (W^{n+k}(A) \cap F^{p_k}(A)) + W^{n+k+1}(A) \\ C' &= (W^{n+k}(A) \cap \bar {F}^{q_k}(A)) + W^{n+k+1}(A) \\ D &= W^{n+k}(A) \\ E &= W^{n+k+1}(A). \end {aligned}
can then be written as
\begin {aligned} B'+C' &= D \\ B' \cap C' &= E. \end {aligned}
We also have, since p_k \leqslant p_i (for i \leqslant k),
B \cap D \subset F^{p_k}(A) \cap W^{n+k}(A) \subset B'
and, since q_k \leqslant q_i (for i \leqslant k),
C \cap D \subset \bar {F}^{q_k}(A) \cap W^{n+k}(A) \subset C'.
The claim (*_{k+1}) and then follows from (*_k) and the following lemma.
677Lemmahodge-theory-ii-1.2.9hodge-theory-ii-1.2.9.xml1.2.9hodge-theory-ii-1.2
Let B, C, B', C', D, and E be sub-objects of A.
Suppose that
\begin {gathered} B'+C' = D \qquad B' \cap C' = E \\ B \cap D \subset B' \qquad C \cap D \subset C'. \end {gathered}
Then
((B+B') \cap (C+C'))/E \xrightarrow { \sim } ((B+D) \cap (C+D))/D.
676Proofhodge-theory-ii-1.2.9
To prove surjectivity, we write
\begin {aligned} ((B+B') \cap (C+C'))+D &= (((B+B') \cap (C+C'))+B')+C' \\ &= ((B+B') \cap (C+C'+B'))+C' \\ &= (B+B'+C') \cap (C+C'+B') \\ &= (B+D) \cap (C+D). \end {aligned}
To prove injectivity, we write
(B+B') \cap (C+C') \cap D = ((B+B') \cap D) \cap ((C+C') \cap D).
Since B' \subset D, we have
\begin {aligned} (B+B') \cap D &= (B \cap D)+B' \\ &= B' \end {aligned}
and similarly
(C+C') \cap D = C'
and
\begin {aligned} (B+B') \cap (C+C') \cap D &= B' \cap C' \\ &= E. \end {aligned}
This finishes the proof of , noting that is equivalent to (*_k) for large k.
683Theoremhodge-theory-ii-1.2.10hodge-theory-ii-1.2.10.xml1.2.10hodge-theory-ii-1.2
Let \mathscr {A} be an abelian category, and provisionally denote by \mathscr {A}' the category of objects of \mathscr {A} endowed with three opposite filtrations W, F, and \bar {F}.
The morphisms in \mathscr {A}' are the morphisms of \mathscr {A} that are compatible with the three filtrations.
\mathscr {A}' is an abelian category.
The kernel (resp. cokernel) of an arrow f \colon A \to B in \mathscr {A}' is the kernel (resp. cokernel) of f in \mathscr {A}, endowed with the filtrations induced by those of A (resp. the quotients of those of B).
Every morphism f \colon A \to B in \mathscr {A}' is strictly compatible with the filtrations W, F, and \bar {F};
the morphism \operatorname {Gr} _W(f) is compatible with the bigradings of \operatorname {Gr} _W(A) and \operatorname {Gr} _W(B);
the morphisms \operatorname {Gr} _F(f) and \operatorname {Gr} _{ \bar {F}}(f) are strictly compatible with the filtration induced by W.
The "forget the filtrations" functors, \operatorname {Gr} _W, \operatorname {Gr} _F, and \operatorname {Gr} _{ \bar {F}}, and
\begin {gathered} \operatorname {Gr} _W \operatorname {Gr} _F \simeq \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _W \simeq \operatorname {Gr} _W \operatorname {Gr} _{ \bar {F}} \end {gathered}
from \mathscr {A}' to \mathscr {A} are exact.
Denote by \sigma _0(p,q) and \sigma _1(p,q) the sequences
\begin {aligned} \sigma _0(p,q) &= (p,q), (p,q), (p,q-1), (p,q-2), (p,q-3), \ldots \\ \sigma _0(p,q) &= (p,q), (p,q), (p-1,q), (p-2,q), (p-3,q), \ldots \end {aligned}
and, with the notation of , set
A_i^{p,q} = A_{ \sigma _i(p,q)} \qquad \text {for } i=0,1.
If f \colon A \to B is compatible with W, F, and \bar {F}, then we have
684Equationhodge-theory-ii-1.2.10.1hodge-theory-ii-1.2.10.1.xml1.2.10.1hodge-theory-ii-1.2 f(A_i^{p,q}) \subset B_i^{p,q} \qquad \text {for }i=0,1. \tag{1.2.10.1}
Claim (iii) then follows from the following lemma:
690Lemmahodge-theory-ii-1.2.11hodge-theory-ii-1.2.11.xml1.2.11hodge-theory-ii-1.2
The A_i^{p,q} give a bigrading of A.
We have
685Equationhodge-theory-ii-1.2.11.1hodge-theory-ii-1.2.11.1.xml1.2.11.1hodge-theory-ii-1.2.11 W^n(A) = \sum _{n+p+q \leqslant 0} A_i^{p,q} \qquad \text {for }i=0,1 \tag{1.2.11.1}
686Equationhodge-theory-ii-1.2.11.2hodge-theory-ii-1.2.11.2.xml1.2.11.2hodge-theory-ii-1.2.11 F^p(A) = \sum _{p' \geqslant p} A_0^{p',q'} \tag{1.2.11.2}
687Equationhodge-theory-ii-1.2.11.3hodge-theory-ii-1.2.11.3.xml1.2.11.3hodge-theory-ii-1.2.11 \bar {F}^q(A) = \sum _{q' \geqslant q} A_1^{p',q'}. \tag{1.2.11.3}
689Proofhodge-theory-ii-1.2.11
By symmetry, it suffices to prove the claims concerning i=0.
Set A_0= \bigoplus A_0^{p,q} and define filtrations W and F on A_0 by the equations of .
The canonical map i from A_0 to A is compatible with the filtrations W and F.
Furthermore, by , \operatorname {Gr} _W(i) is an isomorphism, and induces isomorphisms of graded objects
688Equationhodge-theory-ii-1.2.11.4hodge-theory-ii-1.2.11.4.xml1.2.11.4hodge-theory-ii-1.2.11 \sum _{p+q=n} A_0^{p,q} \xrightarrow { \sim } \operatorname {Gr} _W^{-n}(A) = \sum _{p+q=n} A^{p,q}. \tag{1.2.11.4}
The morphism i is thus an isomorphism, and the A_0^{p,q} give a bigrading of A.
then says that \operatorname {Gr} _W(i) is an isomorphism.
By , \operatorname {Gr} _F \operatorname {Gr} _W(i) is an isomorphism, and thus so too are \operatorname {Gr} _W \operatorname {Gr} _F(i) and \operatorname {Gr} _F(i).
Equation then follows.
691hodge-theory-ii-1.2.12hodge-theory-ii-1.2.12.xml1.2.12hodge-theory-ii-1.2
We now prove .
Let f \colon A \to B in \mathscr {A}' and endow K= \operatorname {Ker} (f) with the filtrations induced by those of A.
By , \operatorname {Gr} _W(K) \hookrightarrow \operatorname {Gr} _W(A);
furthermore, the filtration F (resp. \bar {F}) on K induces on \operatorname {Gr} _W(K) the inverse image filtration of the filtration F on \operatorname {Gr} _W(A).
The sub-object \operatorname {Gr} _W(K) of \operatorname {Gr} _W(A) is then compatible with the bigrading of \operatorname {Gr} _W(A):
\operatorname {Gr} _W(K) = \bigoplus _{p,q}( \operatorname {Gr} _W(K) \cap A^{p,q}).
We thus deduce that
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(K) \hookrightarrow \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A);
the filtrations of W, F, and \bar {F} on K are thus opposite, and K is a kernel of f in \mathscr {A}'.
This, combined with the dual result, proves (ii).
If f is an arrow of \mathscr {A}', then the canonical morphism from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism in \mathscr {A};
by (iii), it is also an isomorphism in \mathscr {A}', which is thus abelian.
The "forget the filtrations" functor is exact by (ii).
The exactness of the other functors in (iv) follows immediately from (ii), (iii), and (i) or (ii) of .
692hodge-theory-ii-1.2.13hodge-theory-ii-1.2.13.xml1.2.13hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with a finite increasing filtration W_ \bullet, and two finite decreasing filtrations F and \bar {F}.
The construction associates to W_ \bullet a decreasing filtration W^ \bullet.
We say that the filtrations W_ \bullet, F, and \bar {F} are opposite if the filtrations W^ \bullet, F, and \bar {F} are, i.e. if, for all n, the filtrations induced by F and \bar {F} on
\operatorname {Gr} _n^W(A) = W_n(A)/W_{n-1}(A)
are n-opposite.
translates trivially to this variation.
1636hodge-theory-ii-1.3hodge-theory-ii-1.3.xmlThe two filtrations lemma1.3hodge-theory-ii-1823hodge-theory-ii-1.3.1hodge-theory-ii-1.3.1.xml1.3.1hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with a filtration F.
The filtration is said to be biregular if it induces a finite filtration on each component of K.
We recall the definition of the terms E_r^{pq}(K,F), or simply E_r^{pq}, of the spectral sequence defined by F.
We set
Z_r^{pq} = \operatorname {Ker} (d \colon F^p(K^{p+q}) \to F^{p+q+1}/F^{p+r}(K^{p+q+1}))
and dually we define B_r^{pq} by the formula
K^{p+q}/B_r^{pq} = \operatorname {Coker} (d \colon F^{p-r+1}(K^{p+q+1}) \to K^{p+q}/F^{p+1}(K^{p+q})).
These formulas still make sense for r= \infty.
We note that the use here of the notation B_r^{pq} is different to that of Godement [G1958].
We have, by definition:
395Equationhodge-theory-ii-1.3.1.1hodge-theory-ii-1.3.1.1.xml1.3.1.1hodge-theory-ii-1.3.1 E_r^{pq} = \operatorname {Im} (Z_r^{pq} \to K^{p+q}/B_r^{pq}) \tag{1.3.1.1}
396Equationhodge-theory-ii-1.3.1.2hodge-theory-ii-1.3.1.2.xml1.3.1.2hodge-theory-ii-1.3.1 = Z_r^{pq}/(B_r^{pq} \cap Z_r^{pq}) \tag{1.3.1.2}
397Equationhodge-theory-ii-1.3.1.3hodge-theory-ii-1.3.1.3.xml1.3.1.3hodge-theory-ii-1.3.1 = \operatorname {Ker} (K^{p+q}/B_r^{pq} \to K^{p+q}/(Z_r^{pq}+B_r^{pq})). \tag{1.3.1.3}
We can thus write
398Equationhodge-theory-ii-1.3.1.4hodge-theory-ii-1.3.1.4.xml1.3.1.4hodge-theory-ii-1.3.1 \begin {aligned} B_r^{p \bullet } \cap Z_r^{p \bullet } & \coloneqq (dF^{p-r+1}+F^{p+1}) \cap (d^{-1}F^{p+r} \cap F^p) \\ &= (dF^{p-r+1} \cap F^p) + (F^{p+1} \cap d^{-1}F^{p+r}) \end {aligned} \tag{1.3.1.4}
since dF^{p-r+1} \subset d^{-1}F^{p+r} and F^{p+1} \subset F^p.
For r< \infty, the E_r form a complex graded by the degree p-r(p+q), and E_{r+1} can be expressed as the cohomology of this complex:
399Equationhodge-theory-ii-1.3.1.5hodge-theory-ii-1.3.1.5.xml1.3.1.5hodge-theory-ii-1.3.1 E_{r+1}^{pq} = \operatorname {H} (E_r^{p-r,q+r-1} \xrightarrow {d_r} E_r^{pq} \xrightarrow {d_r} E_r^{p+r,q-r+1}). \tag{1.3.1.5}
For r=0, we have
400Equationhodge-theory-ii-1.3.1.6hodge-theory-ii-1.3.1.6.xml1.3.1.6hodge-theory-ii-1.3.1 E_0^{ \bullet \bullet } = \operatorname {Gr} _F^ \bullet (K^ \bullet ). \tag{1.3.1.6} 828Propositionhodge-theory-ii-1.3.2hodge-theory-ii-1.3.2.xml1.3.2hodge-theory-ii-1.3
Let K be a complex endowed with a biregular filtration F.
The following conditions are equivalent:
The spectral sequence defined by F degenerates (E_1=E_ \infty).
The morphisms d \colon K^i \to K^{i+1} are strictly compatible with the filtrations.
827Proofhodge-theory-ii-1.3.2
We will prove this in the case where \mathscr {A} is a category of modules.
For fixed p and q, the hypothesis that the arrows d_r with domains E_r^{pq} be zero for r \geqslant1 implies that, if x \in F^p(K^{p+q}) satisfies dx \in F^{p+1}(K^{p+q+1}), then there exists y \in K^{p+q} such that dy=0 and such that x and y have the same image in E_1^{pq}.
Modifying y by a boundary, and setting z=x-y, we then have
\forall x \in F^p(K^{p+q}) \left [ dx \in F^{p+1}(K^{p+q+1}) \implies \exists z \text { s.t. } z \in F^{p+1}(K^{p+q}) \text { and } dz=dx \right ]
or, in other words,
F^{p+1}(K^{p+q+1}) \cap dF^p(K^{p+q}) = dF^{p+1}(K^{p+q}). \tag{1}
If this condition is satisfied for arbitrary p and q, then by induction on r we have
F^{p+r} \cap dF^p = dF^{p+r}
which, for large p+r, can be written as
F^p \cap dK = dF^p. \tag{2}
Claim (2) trivially implies (1), and is equivalent to (ii), which proves the proposition.
831hodge-theory-ii-1.3.3hodge-theory-ii-1.3.3.xml1.3.3hodge-theory-ii-1.3
If (K,F) is a filtered complex, we denote by \operatorname {Dec} (K) the complex K endowed with the shifted filtration
\operatorname {Dec} (F)^p K^n = Z_1^{p+n,-p}.
This filtration is compatible with the differentials:
\begin {aligned} dZ_1^{p+n,-p} & \subset F^{p+n+1}(K^{n+1}) \cap \operatorname {Ker} (d) \\ & \subset Z_ \infty ^{p+n+1,-p} \\ & \subset Z_1^{p+n+1,-p}. \end {aligned}
Since
829Equationhodge-theory-ii-1.3.3.1hodge-theory-ii-1.3.3.1.xml1.3.3.1hodge-theory-ii-1.3.3 \begin {aligned} Z_1^{p+1+n,-p-1} & \subset F^{p+1+n}(K^n) \\ & \subset B_1^{p+n,-p} \\ & \subset Z_1^{p+n,-p} \end {aligned} \tag{1.3.3.1}
the evident arrow from Z_1^{p+n,-p}/Z_1^{p+1+n,-p-1} to Z_1^{p+n,-p}/B_1^{p+n,-p} is a morphism
830Equationhodge-theory-ii-1.3.3.2hodge-theory-ii-1.3.3.2.xml1.3.3.2hodge-theory-ii-1.3.3 u \colon E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,-p}(K). \tag{1.3.3.2} 838Propositionhodge-theory-ii-1.3.4hodge-theory-ii-1.3.4.xml1.3.4hodge-theory-ii-1.3
The morphisms in form a morphism of graded complexes from E_0( \operatorname {Dec} (K)) to E_1(K).
This morphism induces an isomorphism on cohomology.
This morphism induces step-by-step (via ) isomorphisms of graded complexes E_r( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}(K) (for r \geqslant1).
837Proofhodge-theory-ii-1.3.4
Let F' be the filtration on K defined by
{F'}^p(K^n) = \operatorname {Dec} (F)^{p-n}(K^n) = Z_1^{p,n-p}.
We trivially have isomorphisms
836Equationhodge-theory-ii-1.3.4.1hodge-theory-ii-1.3.4.1.xml1.3.4.1hodge-theory-ii-1.3.4 E_r^{p,n-p}( \operatorname {Dec} (K)) = E_{r+1}^{p+n,-p}(K,F') \tag{1.3.4.1}
that are compatible with the d_r and with .
The map u comes from and from the identity map
(K,F') \to (K,F).
This proves (i), and it remains to show that, for r \geqslant2,
E_r^{pq}(K,F') \xrightarrow { \sim } E_r^{pq}(K,F).
We have
\begin {aligned} Z_r^{pq}(K,F') &= Z_r^{pq}(K,F) \qquad \text {for }r \geqslant1 \\ Z_r^{pq}(K,F') \cap B_r^{pq}(K,F') &= Z_r^{pq}(K,F) \cap B_r^{pq}(K,F) \qquad \text {for }r \geqslant2 \end {aligned}
and we can then apply .
839hodge-theory-ii-1.3.5hodge-theory-ii-1.3.5.xml1.3.5hodge-theory-ii-1.3
The construction in is not self-dual.
The dual construction consists of defining
\operatorname {Dec} ^ \bullet (F)^pK^n = B_1^{p+n-1,-p+1}.
We then have morphisms
E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,p}(K) \to E_0^{p,n-p}( \operatorname {Dec} ^ \bullet (K))
and, for r \geqslant1, isomorphisms
E_r^{p,n-p}( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}^{p+n,p}(K) \xrightarrow { \sim } E_{r}^{p,n-p}( \operatorname {Dec} ^ \bullet (K)).
Recall that a morphism of complexes is said to be a quasi-isomorphism if it induces an isomorphism on cohomology.
843Definitionhodge-theory-ii-1.3.6hodge-theory-ii-1.3.6.xml1.3.6hodge-theory-ii-1.3
A morphism f \colon (K,F) \to (K',F') of filtered complexes with biregular filtrations is a filtered quasi-isomorphism if \operatorname {Gr} _F(f) is a quasi-isomorphism, i.e. if the E_1^{pq}(f) are isomorphisms.
A morphism f \colon (K,F,W) \to (K,F',W') of biregular bifiltered complexes is a bifiltered quasi-isomorphism if \operatorname {Gr} _F \operatorname {Gr} _W(f) is a quasi-isomorphism.
844hodge-theory-ii-1.3.7hodge-theory-ii-1.3.7.xml1.3.7hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with two filtrations F and W.
Let E_r^{pq} be the spectral sequence defined by W.
The filtration F induces on E_r^{pq} various filtrations, which we will compare.
845hodge-theory-ii-1.3.8hodge-theory-ii-1.3.8.xml1.3.8hodge-theory-ii-1.3 identifies E_r^{pq} with a quotient of a sub-object of K^{p+q}.
The E_r^{pq} term is thusly given by endowing with a filtration F_d induced by F, called the first direct filtration.
846hodge-theory-ii-1.3.9hodge-theory-ii-1.3.9.xml1.3.9hodge-theory-ii-1.3
Dually, identifies E_r^{pq} with a sub-object of a quotient of K^{p+q}, whence a new filtration F_{d^*} induced by F, called the second direct filtration.
847Lemmahodge-theory-ii-1.3.10hodge-theory-ii-1.3.10.xml1.3.10hodge-theory-ii-1.3
On E_0 and E_1, we have F_d=F_{d^*}.
651Proofhodge-theory-ii-1.3.10
For r=0,1, we have B_r^{pq} \subset Z_r^{pq}, and we apply .
851hodge-theory-ii-1.3.11hodge-theory-ii-1.3.11.xml1.3.11hodge-theory-ii-1.3 identifies E_{r+1}^{pq} with a quotient of a sub-object of E_r^{pq}.
We define the recurrent filtration F_r on the E_r^{pq} by the conditions
On E_0^{pq}, F_r=F_d=F_{d^*}.
On E_{r+1}^{pq}, the recurrent filtration is that induced by the recurrent filtration of E_r^{pq}.
852hodge-theory-ii-1.3.12hodge-theory-ii-1.3.12.xml1.3.12hodge-theory-ii-1.3
Definitions and still make sense for r= \infty.
If the filtration on K is biregular, then the direct filtrations on E_ \infty ^{pq} coincide with those on E_r^{pq}=E_ \infty ^{pq} for large enough r, and we define the recurrent filtration on E_ \infty ^{pq} as agreeing with that on E_r^{pq} for large enough r.
The filtrations F and W each induce a filtration on H^ \bullet (K), and E_ \infty ^{ \bullet \bullet }= \operatorname {Gr} _W^ \bullet ( \operatorname {H} ^ \bullet (K)).
The filtration F on \operatorname {H} ^ \bullet (K) then induces on E_ \infty ^{pq} a new filtration.
859Propositionhodge-theory-ii-1.3.13hodge-theory-ii-1.3.13.xml1.3.13hodge-theory-ii-1.3
For the first direct filtration, the morphisms d_r are compatible with the filtrations.
If E_{r+1}^{pq} is considered as a quotient of a sub-object of E_r^{pq}, then the first direct filtration on E_{r+1}^{pq} is finer than the filtration F' induced by the first direct filtration on E_r^{pq}Y we have F_d(E_{r+1}^{pq}) \subset F'(E_{r+1}^{pq}).
Dually, the morphisms d_r are compatible with the second direct filtration, and the second direct filtration on E_{r+1}^{pq} is less fine than the filtration induced by that of E_r^{pq}.
F_d(E_r^{pq}) \subset F_r(E_r^{pq}) \subset F_{d^*}(E_r^{pq}).
On E_ \infty ^{pq}, the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K) is finer than the first direct filtration and less fine than the second.
858Proofhodge-theory-ii-1.3.13
Claim (i) is evident, (ii) is its dual, and (iii) follows by induction.
The first claim of (iv) is easy to verify, and the second is its dual.
860hodge-theory-ii-1.3.14hodge-theory-ii-1.3.14.xml1.3.14hodge-theory-ii-1.3
We denote by \operatorname {Dec} (K) (resp. \operatorname {Dec} ^ \bullet (K)) the complex K endowed with the filtrations \operatorname {Dec} (W) and F (resp. \operatorname {Dec} ^ \bullet (W) and F).
It is clear by that the isomorphism in sends the first direct filtration on E_r( \operatorname {Dec} (K)) to the second direct filtration on E_{r+1}(K) (for r \geqslant1).
The dual isomorphism sends the second direct filtration on E_r( \operatorname {Dec} ^ \bullet (K)) to the second direct filtration on E_{r+1}(K).
865Lemmahodge-theory-ii-1.3.15hodge-theory-ii-1.3.15.xml1.3.15hodge-theory-ii-1.3
If the filtration F is biregular, and if, on the \operatorname {Gr} _W^p(K), the morphisms d are strictly compatible with the filtration induced by F, then
The morphism of graded complexes filtered by F
u \colon \operatorname {Gr} _{ \operatorname {Dec} (W)}(K) \to E_1(K,W)
is a filtered quasi-isomorphism.
Dually, the morphism in
u \colon E_1(K,W) \to \operatorname {Gr} _{ \operatorname {Dec} ^ \bullet (W)}(K)
is a filtered quasi-isomorphism.
864Proofhodge-theory-ii-1.3.15
It suffices, by duality, to prove (i).
By and , the complex E_1(K,W) filtered by F is a quotient of the filtered complex \operatorname {Gr} _{ \operatorname {Dec} (W)}(K).
Let U be the filtered complex given by the kernel, which is acyclic by (ii) of .
The long exact sequence in cohomology associated to the exact sequence of complexes
0 \to \operatorname {Gr} _F(U) \to \operatorname {Gr} _F( \operatorname {Gr} _{ \operatorname {Dec} (W)}(K)) \to \operatorname {Gr} _F(E_1(K,W)) \to 0
shows that u is a filtered quasi-isomorphism if and only if \operatorname {Gr} _F(U) is an acyclic complex.
By , and since U is acyclic, this reduces to asking that the differentials of U be strictly compatible with the filtration F.
From we obtain that U is the sum over p of the complexes
(U^p)^n = B_1^{p+n,-p}/Z_1^{p+1+n,-p-1}
endowed with the filtration induced by F.
Each differential d of each of the complexes U^p fits into a commutative diagram of filtered objects of the following type, where, for simplicity, we have omitted the total or complementary degree:
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
B^p/Z^{p+1}
\ar [rrr,"d"]
\ar [dr]
&&& B^{p+1}/Z^{p+1}
\ar [dd]
\\ & \operatorname {Coim} (d)
\ar [r,"(5)"]
& \operatorname {Im} (d)
\ar [ur] \ar [dr]
\\ W^{p+1}/Z^{p+1}
\ar [uu]
\ar [ur]
\ar [rrr,near start,swap,"(3)"]
\ar [white,urr,near end,swap," \color {black}(4)"]
&&& B^{p+1}/W^{p+2}
\ar [d]
\\ W^{p+1}/W^{p+2}
\ar [u]
\ar [rrr,near start,swap,"(1)"]
\ar [white,urrr," \color {black}(2)"description]
&&& W^{p+1}/W^{p+2}
\end {tikzcd}
By hypothesis, the morphism (1) is strict.
Since the square (2) is exactly the canonical decomposition of (1), the arrow (3) is a filtered isomorphism.
The arrows of the trapezium (4) are isomorphisms;
they are thus filtered isomorphisms, since (3) is a filtered isomorphism.
The fact that (5) is a filtered isomorphism implies that d is strict.
This proves the lemma.
867Theoremhodge-theory-ii-1.3.16hodge-theory-ii-1.3.16.xml1.3.16hodge-theory-ii-1.3
Let K be a complex endowed with two filtrations, W and F, with the filtration F biregular.
Let r_0 \geqslant0 be an integer, and suppose that, for 0 \leqslant r<r_0, the differentials of the graded complex E_r(K,W) are strictly compatible with the filtration F.
Then, for r \leqslant r_0+1, F_d=F_r=F_{d^*} on E_r^{pq}.
866Proofhodge-theory-ii-1.3.16
We will prove the theorem by induction on r_0.
For r_0=0, the hypothesis is empty, and we apply and (iii) of .
For r_0 \geqslant1, by the inductive hypothesis, we have F_d=F_r=F_{d^*} on E_r^{pq} for r \leqslant r_0.
By , the morphism u \colon E_0( \operatorname {Dec} (K)) \to E_1(K) is a filtered quasi-isomorphism.
It thus induces a filtered isomorphism from \operatorname {H} ^ \bullet ( \operatorname {Dec} (K)) to \operatorname {H} ^ \bullet (E_1(K)):
u \colon (E_1( \operatorname {Dec} (K)),F_r) \xrightarrow { \sim } (E_2(K),F_r).
Step-by-step, we thus deduce that the canonical isomorphism from E_s( \operatorname {Dec} (K)) to E_{s+1} (for s \geqslant1) is a filtered isomorphism for the recurrent filtration.
On E_1( \operatorname {Dec} (K)), F_r=F_d (), and we already know () that u' is a filtered isomorphism
u' \colon (E_1( \operatorname {Dec} (K)),F_d) \xrightarrow { \sim } (E_2(K),F_d).
On E_2(K), we thus have F_d=F_r.
This, combined with the dual result, proves the theorem for r_0=1.
Suppose that r_0 \geqslant2.
Then the arrows d_1 of E_1(K) are strictly compatible with the filtrations, and thus so too are the arrows d_0 of E_0( \operatorname {Dec} (K)) (indeed, u induces an isomorphism of spectral sequences, and we apply the criterion from ).
For 0<s<r_0-1, the isomorphism (E_s( \operatorname {Dec} (K)),F_r) \cong (E_{s+1}(K),F_r) shows that the d_s are strictly compatible with the recurrent filtrations.
By the induction hypothesis, we thus have F_d=F_r on E_s( \operatorname {Dec} (K)) for s \leqslant s_0.
The isomorphism (E_s( \operatorname {Dec} (K)),F_d) \cong (E_{s+1}(K),F_d) () then shows that F_d=F_r on E_r(K) for r \leqslant r_0+1.
This, combined with the dual result, proves the theorem.
869Corollaryhodge-theory-ii-1.3.17hodge-theory-ii-1.3.17.xml1.3.17hodge-theory-ii-1.3
Under the general hypotheses of , suppose that, for all r, the differentials d_r are strictly compatible with the recurrent filtrations on the E_r.
Then, on E_ \infty, the filtrations F_d, F_r, and F_{d^*} agree, and coincide with the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K).
868Proofhodge-theory-ii-1.3.17
This follows immediately from and (iv) of .
1637hodge-theory-ii-1.4hodge-theory-ii-1.4.xmlHypercohomology of filtered complexes1.4hodge-theory-ii-1
In this section, we recall some standard constructions in hypercohomology.
We do not use the language of derived categories, which would be more natural here.
Throughout this entire section, by "complex" we mean "bounded-below complex".992hodge-theory-ii-1.4.1hodge-theory-ii-1.4.1.xml1.4.1hodge-theory-ii-1.4
Let T be a left-exact functor from an abelian category \mathscr {A} to an abelian category \mathscr {B}.
Suppose that every object of \mathscr {A} injects into an injective object;
the derived functors \mathrm {R} ^iT \colon \mathscr {A} \to \mathscr {B} are then defined.
An object A of \mathscr {A} is said to be *acyclic* for T if \mathrm {R} ^iT(A)=0 for i>0.
994hodge-theory-ii-1.4.2hodge-theory-ii-1.4.2.xml1.4.2hodge-theory-ii-1.4
Let (A,F) be a filtered object with finite filtration, and TF the filtration of TA by its sub-objects TF^p(A) (these are sub-objects since T is left exact).
If \operatorname {Gr} _F(A) is T-acyclic, then the F^p(A) are T-acyclic as successive extensions of T-acyclic objects.
The image under T of the sequence
0 \to F^{p+1}(A) \to F^p(A) \to \operatorname {Gr} ^p(A) \to 0
is thus exact, and
993Equationhodge-theory-ii-1.4.2.1hodge-theory-ii-1.4.2.1.xml1.4.2.1hodge-theory-ii-1.4.2 \operatorname {Gr} _{FT}TA \xrightarrow { \sim } T \operatorname {Gr} _FA. \tag{1.4.2.1} 995hodge-theory-ii-1.4.3hodge-theory-ii-1.4.3.xml1.4.3hodge-theory-ii-1.4
Let A be an object endowed with finite filtrations F and W such that \operatorname {Gr} _F \operatorname {Gr} _W A are T-acyclic.
The objects \operatorname {Gr} _FA and \operatorname {Gr} _WA are then T-acyclic, as well as the F^q(A) \cap W^p(A).
The sequences
0 \to T(F^q \cap W^{p+1}) \to T(F^q \cap W^p) \to T((F^q \cap W^p)/(F^q \cap W^{p+1})) \to 0
are thus exact, and T(F^q( \operatorname {Gr} _W^p(A))) is the image in T( \operatorname {Gr} _W^p(A)) of T(F^p \cap W^q).
The diagram
\begin {CD} T(F^q \cap W^p) @>>> T(F^q \operatorname {Gr} _W^pA) @>>> T \operatorname {Gr} _W^pA \\ @V{ \cong }VV @. @VV{ \cong }V \\ TF^q \cap TW^p @= TF^q \cap TW^p @>>> \operatorname {Gr} _{TW}^pTA \end {CD}
then shows that the isomorphism in relative to W sends the filtration \operatorname {Gr} _{TW}(TF) to the filtration T( \operatorname {Gr} _W(F)).
999hodge-theory-ii-1.4.4hodge-theory-ii-1.4.4.xml1.4.4hodge-theory-ii-1.4
Let K be a complex of objects of \mathscr {A}.
The hypercohomology objects \mathrm {R} ^iT(K) are calculated as follows:
We choose a quasi-isomorphism i \colon K \to K such that the components of K' are acyclic for T.
For example, we can take K' to be the simple complex associated to an injective Cartan–Eilenberg resolution of K.
We set
\mathrm {R} ^iT(K) = \operatorname {H} ^i(T(K')).
We can show that \mathrm {R} ^iT(K) does not depend on the choice of K', but depends functorially on K, and that a quasi-isomorphism f \colon K_1 \to K_2 induces *isomorphisms*
\mathrm {R} ^iT(f) \colon \mathrm {R} ^iT(K_1) \to \mathrm {R} ^iT(K_2). 1000hodge-theory-ii-1.4.5hodge-theory-ii-1.4.5.xml1.4.5hodge-theory-ii-1.4
Let F be a biregular filtration of K.
A {#T}-acyclic filtered resolution of K is a filtered quasi-isomorphism i \colon K \to K' from K to a filtered biregular complex such that the \operatorname {Gr} ^p({K'}^n) are acyclic for T.
If K' is such a resolution, then the {K'}^n are acyclic for T, and the filtered complex (cf. ) T(K') defines a spectral sequence
E_1^{pq} = \mathrm {R} ^{p+q}T( \operatorname {Gr} ^p(K)) \Rightarrow \mathrm {R} ^{p+q}T(K).
This is independent of the choice of K'.
We call this the hypercohomology spectral sequence of the filtered complex K.
It depends functorially on K, and a filtered quasi-isomorphism induces an isomorphism of spectral sequences.
The differentials d_1 of this spectral sequence are the connection morphisms defined by the short exact sequences
0 \to \operatorname {Gr} ^{p+1}K \to F^pK/F^{p+2}K \to \operatorname {Gr} ^pK \to 0. 1001hodge-theory-ii-1.4.6hodge-theory-ii-1.4.6.xml1.4.6hodge-theory-ii-1.4
Let K be a complex.
We denote by \tau _{ \leqslant p}(K) the following subcomplex:
\tau _{ \leqslant p}(K)^n = \begin {cases} K^n & \text {for }n<p \\ \operatorname {Ker} (d) & \text {for }n=p \\ 0 & \text {for }n>p. \end {cases}
The filtration, said to be canonical, of K by the \tau _{ \leqslant p}(K) is induced by shifting the trivial filtration G for which G^0(K)=K and G^1(K)=0.
We have, for the canonical filtration,
\begin {aligned} E_1^{pq} = 0 & \qquad \text {if }p+q \neq-p \\ \\ H^{-p} & \qquad \text {if }p+q=-p. \end {aligned}
A quasi-isomorphism f \colon K \to K' is automatically a filtered quasi-isomorphism for the canonical filtrations.
1002hodge-theory-ii-1.4.7hodge-theory-ii-1.4.7.xml1.4.7hodge-theory-ii-1.4
The subcomplexes \sigma _{ \geqslant p}(K) of K
\sigma _{ \geqslant p}(K)^n = \begin {cases} 0 & \text {if }n<p \\ K^n & \text {if }n \geqslant p \end {cases}
define a biregular filtration, called the stupid filtration of K.
The hypercohomology spectral sequences attached to the stupid or canonical filtrations of K are the two hypercohomology spectral sequences of K.
1003Examplehodge-theory-ii-1.4.8hodge-theory-ii-1.4.8.xml1.4.8hodge-theory-ii-1.4
Let f \colon X \to Y be a continuous map between topological spaces, and let \mathscr {F} be an abelian sheaf on X.
Let \mathscr {F}^ \bullet be a resolution of \mathscr {F} by f_*-acyclic sheaves.
We have \mathrm {R} ^i f_* \mathscr {F} \simeq \operatorname { \mathscr {H}} ^i(f_* \mathscr {F}^ \bullet ).
We take the functor T to be the functor \Gamma (Y,-).
The hypercohomology spectral sequence of the complex f_* \mathscr {F}^ \bullet endowed with its canonical filtration
E_1^{pq} = \operatorname {H} ^{2p+q}(Y, \mathrm {R} ^{-p}f_* \mathscr {F}) \Rightarrow \operatorname {H} ^{p+q}(X, \mathscr {F})
is exactly, up to the renumbering E_r^{pq} \mapsto E_{r+1}^{2p+q,-p}, the Leray spectral sequence for f and \mathscr {F}.
1010hodge-theory-ii-1.4.9hodge-theory-ii-1.4.9.xml1.4.9hodge-theory-ii-1.4
Let (K,W,F) be a biregular bifiltered complex.
To this complex, we associate:
A spectral sequence
{}_W E_1^{p,n-p} = \operatorname {H} ^n( \operatorname {Gr} _W^p(K)) \Rightarrow \operatorname {H} ^n(K)
with differentials {}_W d_1 being the connecting morphisms induced by the short exact sequences
0 \to \operatorname {Gr} _W^{p+1}(K) \to W^p(K)/W^{p+2}(K) \to \operatorname {Gr} _W^p(K) \to 0;
An analogous spectral sequence for the filtration F;
Exact squares
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& 0 \dar
& 0 \dar
& 0 \dar
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^{q+1}K \rar \dar
& F^p/F^{p+2}( \operatorname {Gr} _W^{q+1}K) \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1}K \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1}(W^q/W^{q+2}(K)) \rar \dar
& F^p/F^{p+2}(W^q/W^{q+2}(K)) \rar \dar
& \operatorname {Gr} _F^p(W^q/W^{q+2}(K)) \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \rar \dar
& F^p/F^{p+2} \operatorname {Gr} _W^q K \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^q K \rar \dar
& 0
\\ & 0
& 0
& 0
\end {tikzcd}
The exterior rows and columns of this square define connection morphisms
\begin {aligned} {}_{F,W}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \\ {}_{W,F}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1} K. \end {aligned}
These morphisms satisfy
\begin {gathered} {}_{FW}d_1 \circ {}_{WF}d_1 + {}_{WF}d_1 \circ {}_{FW}d_1 = 0 \\ {}_{FW}d_1^2 = 0 \\ {}_{WF}d_1^2 = 0 \end {gathered}
The morphisms {}_{FW}d_1 are the morphisms d_1 of the spectral sequences E^{(q)}, with the E_1^{(q)p,n-p} term equal to
1008Equationhodge-theory-ii-1.4.9.1hodge-theory-ii-1.4.9.1.xml1.4.9.1hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _W^q K) = {}_WE_1^{q,n-q} \tag{1.4.9.1}
defined by the filtered complex \operatorname {Gr} _W^q(K).
This spectral sequence abuts to the filtration induced by F on \operatorname {H} ^ \bullet \operatorname {Gr} _W^q K.
Similarly, the {}_{WF}d_1 are the d_1 of spectral sequences with the same initial terms
1009Equationhodge-theory-ii-1.4.9.2hodge-theory-ii-1.4.9.2.xml1.4.9.2hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _F^p K). \tag{1.4.9.1}
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& \operatorname {H} ^n \operatorname {Gr} _W^q K
\ar [Rightarrow,dr,"{}_Wd_1"]
\\ \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K
\ar [Rightarrow,ur,"{}_{F,W}d_1"]
\ar [Rightarrow,dr,swap,"{}_{W,F}d_1"]
&& \operatorname {H} ^n K
\\ & \operatorname {H} ^n \operatorname {Gr} _F^p K
\ar [Rightarrow,ur,swap,"{}_Fd_1"]
\end {tikzcd}
These constructions are symmetric in F and W via the isomorphism
\operatorname {Gr} _F^p \operatorname {Gr} _W^q \sim \operatorname {Gr} _W^q \operatorname {Gr} _F^p. 1011hodge-theory-ii-1.4.10hodge-theory-ii-1.4.10.xml1.4.10hodge-theory-ii-1.4
We can also interpret the {}_{W,F}d_1 are the initial morphisms of a morphism of spectral sequences , abutting to {}_Wd_1.
Indeed, let C^q be the cone of the morphism
W^q(K)/W^{q+2}(K) \to \operatorname {Gr} _W^q(K).
In the diagram
\Sigma \colon \operatorname {Gr} _W^{q+1}(K)[1] \xrightarrow {u} C^q \xleftarrow {i} \operatorname {Gr} _W^q(K)
the morphism u is a quasi-isomorphism, and we have
{}_Wd_1 = \operatorname {H} (u)^{-1} \circ \operatorname {H} (i).
In fact, u is even a filtered (for F) quasi-isomorphism, and the above construction defines a morphism from the spectral sequence defined by ( \operatorname {Gr} _W^q(K),F) to that defined by ( \operatorname {Gr} _W^{q+1}(K)[1],F), and it abuts to {}_Wd_1.
The initial term of this morphism, induced by \operatorname {Gr} _F( \Sigma ) is exactly {}_{W,F}d_1.
1012hodge-theory-ii-1.4.11hodge-theory-ii-1.4.11.xml1.4.11hodge-theory-ii-1.4
These constructions pass as they are to hypercohomology.
Let K be a complex endowed with two biregular filtrations F and W.
A bifiltered T-acyclic resolution of K is a bifiltered quasi-isomorphism i \colon K \to K' such that the \operatorname {Gr} _F^p \operatorname {Gr} _W^n({K'}^m) are T-acyclic.
Such a morphism always exists.
In the particular case where \mathscr {A} is the category of sheaves of A-modules on a topological space X, and where T is the functor \Gamma from \mathscr {A} to the category of A-modules, then an example of a bifiltered T-acyclic resolution of K is the simple complex associated to the double complex given by the Godement resolution \mathscr {C}^ \bullet (K) of K, filtered by the \mathscr {C}^ \bullet (F^p(K)) and the \mathscr {C}^ \bullet (W^n(K)).
Since \mathscr {C}^ \bullet is exact, we have
\operatorname {Gr} _F \operatorname {Gr} _W( \mathscr {C}^ \bullet (K)) \simeq \mathscr {C}^ \bullet ( \operatorname {Gr} _F \operatorname {Gr} _W(K)).
We will not have need here of any other case.
If K' is a bifiltered T-acyclic resolution of K, then the complex TK' is filtered by the TF^pK' and the TW^qK' ().
Furthermore, \operatorname {Gr} _W^n(K') is a T-acyclic filtered (for F) resolution of \operatorname {Gr} _F^n(K), and \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K') is a T-acyclic resolution of \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K), and
\begin {gathered} T \operatorname {Gr} _F K' \approx \operatorname {Gr} _F TK' \qquad \text {as }W \text {-filtered complexes} \\ T \operatorname {Gr} _W K' \approx \operatorname {Gr} _W TK' \qquad \text {as }F \text {-filtered complexes} \\ T \operatorname {Gr} _F \operatorname {Gr} _W K' \approx \operatorname {Gr} _F \operatorname {Gr} _W TK'. \end {gathered} 1013Lemmahodge-theory-ii-1.4.12hodge-theory-ii-1.4.12.xml1.4.12hodge-theory-ii-1.4
Under the hypotheses of :
The initial terms of the hypercohomology spectral sequences
{}_WE_1^{q,n-q} = \mathrm {R} ^nT( \operatorname {Gr} _W^qK) \Rightarrow \mathrm {R} ^nT(K) \tag{1}
{}_FE_1^{p,n-p} = \mathrm {R} ^nT( \operatorname {Gr} _F^pK) \Rightarrow \mathrm {R} ^nT(K) \tag{2}
are abutments of the hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _W^qK and \operatorname {Gr} _F^pK, with E_1 pages given by
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_WE_1^{q,n-q} \qquad \text {for fixed }q \tag{3}
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_FE_1^{p,n-p} \qquad \text {for fixed }p \tag{4}
The filtration of {}_WE_1^{p,n-p}, abutting to the spectral sequence (3), is the filtration of {}_WE_1^{p,n-p}(TK') induced by the filtration F of TK'.
For the differentials of the complexes \operatorname {Gr} _W^n(T(K')) to be strictly compatible with the filtration F, it is necessary and sufficient that the hypercohomology spectral sequences (3) degenerate on the E_1 page.
The morphisms d_1 of the spectral sequence (4) are the initial terms of the degree-1 morphisms of the spectral sequences (3) that abut to the morphisms d_1 of the spectral sequence (1).
895Proofhodge-theory-ii-1.4.12
Claims (i) and (iv) follow from and applied to TK' via the isomorphisms .
Claim (ii) is then trivial by the definition of the recurrent filtration F (identical to the discrete filtrations by and (iii) of ), and claim (iii) follows from .
1695hodge-theory-ihodge-theory-i.xmlHodge Theory I
P. Deligne.
"Théorie de Hodge I".
Actes du Congrès intern. math. 1 (1970) pp. 425–430.
publications.ias.edu/node/359
We intend to give a heuristic dictionary between statements in l-adic cohomology and statements in Hodge theory.
This dictionary has as its most notable sources [S1960] and the conjectural theory of Grothendieck motives [D1969].
Up until now, it has mainly served to formulate conjectures in Hodge theory, and it has sometimes even suggested a proof.
289hodge-theory-i-1hodge-theory-i-1.xml1hodge-theory-i287Definitionhodge-theory-i-definition-1.1hodge-theory-i-definition-1.1.xml1.1hodge-theory-i-1
A mixed Hodge structure H consists of
a \mathbb {Z}-module H_ \mathbb {Z} of finite type (the "integer lattice");
a finite increasing filtration W of H_ \mathbb {Q} =H_ \mathbb {Z} \otimes \mathbb {Q} (the "weight filtration");
a finite decreasing filtration F of H_ \mathbb {C} =H_ \mathbb {Z} \otimes \mathbb {C} (the "Hodge filtration").
This data is subject to the following condition:
there exists a (unique) bigradation of \operatorname {Gr} _W(H_ \mathbb {C} ) by subspaces H^{p,q} such that
\operatorname {Gr} _W^n(H_ \mathbb {C} )= \bigoplus _{p+q=n}H^{p,q}
the filtration F induces on \operatorname {Gr} _W(H_ \mathbb {C} ) the filtration
\operatorname {Gr} _W(F)^p = \bigoplus _{p' \geq p} H^{p',q'}
\overline {H^{pq}}=H^{qp}.
A morphism f \colon H \to H' is a homomorphism f_ \mathbb {Z} \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} such that f_ \mathbb {Q} \colon H_ \mathbb {Q} \to H'_ \mathbb {Q} and f_ \mathbb {C} \colon H_ \mathbb {C} \to H'_ \mathbb {C} are compatible with the filtrations W and F (respectively).
288Equationhodge-theory-i-1.2hodge-theory-i-1.2.xml1.2hodge-theory-i-1
The Hodge numbers of H are the integers
h^{pq} = \dim H^{pq} = h^{qp}. \tag{1.2}
We say that H is pure of weight n if h^{pq}=0 for p+q \neq n (i.e. if \operatorname {Gr} _W^i(H)=0 for i \neq n).
We also say that H is a Hodge structure of weight n.
The Tate Hodge structure \mathbb {Z} (1) is the Hodge structure of weight -2, of pure type (-1,-1), for which \mathbb {Z} (1)_ \mathbb {C} = \mathbb {C} and \mathbb {Z} (1)_ \mathbb {Z} = 2 \pi i \mathbb {Z} = \operatorname {Ker} ( \exp \colon \mathbb {C} \to \mathbb {C} ^*) \subset \mathbb {C}.
We set \mathbb {Z} (n)= \mathbb {Z} (1)^{ \otimes n}.
We can show that mixed Hodge structures form an abelian category.
If f \colon H \to H' is a morphism, then f_ \mathbb {Q} and f_ \mathbb {C} are strictly compatible with the filtrations W and F (cf. [Hodge Theory II, 2.3.5]).
291hodge-theory-i-2hodge-theory-i-2.xml2hodge-theory-i
Let A be a normal integral ring of finite type over \mathbb {Z}, with field of fractions K, and \overline {K} an algebraic closure of K.
Let K_ \mathrm {nr} be the largest sub-extension of \overline {K} that is unramified at each prime ideal of A.
We know that, or we set,
\pi _1( \operatorname {Spec} (A), \overline {K}) = \operatorname {Gal} (K_ \mathrm {nr}/K).
For every closed point x of \operatorname {Spec} (A), defined by some maximal ideal m_x of A, the residue field k_x=A/m_x is finite;
the point x defines a conjugation class of "Frobenius substitutions" \varphi _x \in \pi _1( \operatorname {Spec} (A), \overline {K}).
We set q_x= \# k_x and F_x= \varphi _x^{-1}.
Let K be a field of finite type over the prime field of characteristic p, let \overline {K} be an algebraic closure of K, let l be a prime number \neq p, and let H be a \mathbb {Z} _l-module (or a \mathbb {Q} _l-module) of finite type endowed with a continuous action \rho of \operatorname {Gal} ( \overline {K}/K).
We will still suppose in what follows that there exists an A as above, with l invertible in A, and such that \rho factors through \pi _1( \operatorname {Spec} (A), \overline {K}) = \operatorname {Gal} (K_ \mathrm {nr}/K).
We say that H is pure of weight n if, for every closed point x of an non-empty open subset of \operatorname {Spec} (A), the eigenvalues \alpha of F_x acting on H are algebraic integers whose complex conjugates are all of absolute value | \alpha |=q_x^{n/2}.
290Principlehodge-theory-i-principle-2.1hodge-theory-i-principle-2.1.xml2.1hodge-theory-i-2
If the Galois module H "comes from algebraic geometry", then there exists a (unique) increasing filtration W on H_{ \mathbb {Q} _l}=H \otimes _{ \mathbb {Z} _l} \mathbb {Q} _l (the "weight filtration") that is Galois invariant and such that \operatorname {Gr} _n^W(H) is pure of weight n.
We can also further suppose that \operatorname {Gr} _n^W(H) is semi-simple.
When we have a resolution of singularities we can often give a conjectural definition of W, whose validity follows from the Weil conjectures [W1949] (cf. ).
Let \mu be the subgroup of \overline {K}^* given by the roots of unity.
The Tate module \mathbb {Z} _l(1), defined by
\mathbb {Z} _l(1) = \operatorname {Hom} ( \mathbb {Q} _l/ \mathbb {Z} _l, \mu )
is pure of weight -2.
We set \mathbb {Z} _l(n)= \mathbb {Z} _l(1)^ \otimes n.
It is trivial that every morphism f \colon H \to H' is strictly compatible with the weight filtration.
agrees with the fact that every extension of \mathbb {G}_m ("weight -2") by an abelian variety ("weight -1>-2") is trivial.
306hodge-theory-i-3hodge-theory-i-3.xml3hodge-theory-i305Translationhodge-theory-i-3-translationhodge-theory-i-3-translation.xmlhodge-theory-i-3
The Galois modules that appear in l-adic cohomology have, as analogues, over \mathbb {C}, mixed Hodge structures.
We further have the dictionary:
pure module of weight n
Hodge structure of weight n
weight filtration
weight filtration
Galois-compatible homomorphism
morphism
Tate module \mathbb {Z} _l(1)
Tate Hodge structure \mathbb {Z} (1)
307hodge-theory-i-4hodge-theory-i-4.xml4hodge-theory-i
Let X be a complex algebraic variety (i.e. a scheme of finite type over \mathbb {C} that we assume to be separated).
Then there exists a subfield K of \mathbb {C}, of finite type over \mathbb {Q}, such that X can be defined over K (i.e. it comes from an extension of scalars of K to \mathbb {C} applied to a K-scheme X').
Let \overline {K} be the algebraic closure of K in \mathbb {C}.
The Galois group \operatorname {Gal} ( \overline {K}/K) then acts on the l-adic cohomology groups H^ \bullet (X, \mathbb {Z} _l);
we have
H^ \bullet (X( \mathbb {C} ), \mathbb {Z} ) \otimes \mathbb {Z} _l = H^ \bullet (X, \mathbb {Z} _l) = H^ \bullet (X'_{ \overline {K}}, \mathbb {Z} _l).
By , we should expect for the cohomology groups H^n(X( \mathbb {C} ), \mathbb {Z} ) to carry natural mixed Hodge structures.
This is what we can prove (see [Hodge Theory II, 3.2.5] in the case where X is smooth; the proof is algebraic, using classical Hodge theory [W1958]).
For X projective and smooth, the Weil conjectures imply that H^n(X, \mathbb {Z} _l) is pure of weight n, while classical Hodge theory endows H^n(X, \mathbb {Z} ) with a Hodge structure of weight n.
For every morphism f \colon X \to Y, and for K large enough, f^ \bullet \colon H^ \bullet (Y, \mathbb {Z} _l) \to H^ \bullet (X, \mathbb {Z} _l) Galois-commutes (by structure transport);
similarly, f^ \bullet \colon H^ \bullet (Y, \mathbb {Z} ) \to H^ \bullet (X, \mathbb {Z} ) is a morphism of mixed Hodge structures.
For X smooth, the cohomology class Z in H^{2n}(X, \mathbb {Z} _l(n)) of an algebraic cycle of codimension n defined over K is Galois invariant, i.e. it defines
c(Z) \in \operatorname {Hom} _{ \operatorname {Gal} }( \mathbb {Z} _l(-n),H^{2n}(X, \mathbb {Z} _l)).
Similarly, the cohomology class c(Z) \in H^{2n}(X( \mathbb {C} ), \mathbb {Z} ) is purely of type (n,n), i.e. it corresponds to
c(Z) \in \operatorname {Hom} _{ \mathrm {H.M.}}( \mathbb {Z} (-n),H^{2n}(X( \mathbb {C} ), \mathbb {Z} )). 309hodge-theory-i-5hodge-theory-i-5.xml5hodge-theory-i
If f \colon H \to H' is a Galois-compatible morphism between \mathbb {Q} _l-vector spaces of different weights, then f=0.
Similarly, if f \colon H \to H' is a morphism of pure mixed Hodge structures of different weights, then f is torsion.
A more useful remark is
308Scholiumhodge-theory-i-5-scholiumhodge-theory-i-5-scholium.xmlhodge-theory-i-5
Let H and H' be Hodge structures of weight n and n' (respectively), with n>n'.
Let f \colon H_ \mathbb {Q} \to H'_ \mathbb {Q} be a homomorphism such that f \colon H_ \mathbb {C} \to H'_ \mathbb {C} respects F.
Then f=0.
313hodge-theory-i-6hodge-theory-i-6.xml6hodge-theory-i
Let X be a smooth projective variety over \mathbb {C}, let D= \sum _1^n D_i be a normal crossing divisor in X, with D_i all smooth divisors, and let j be the inclusion of U=X \setminus D into X.
For Q \subset [1,n], we set D_q= \bigcap _{i \in Q}D_i.
In l-adic cohomology, we canonically have
310Equationhodge-theory-i-6.1hodge-theory-i-6.1.xml6.1hodge-theory-i-6 R^q j_* \mathbb {Z} _l = \bigoplus _{ \# Q=q} \mathbb {Z} _l(-q)_{D_Q} \tag{6.1}
and the Leray spectral sequence for j is of the form
311Equationhodge-theory-i-6.2hodge-theory-i-6.2.xml6.2hodge-theory-i-6 E_2^{pq} = \bigoplus _{ \# Q=q} H^p(D_Q, \mathbb {Q} _l) \otimes \mathbb {Z} _l(-q) \Rightarrow H^{p+q}(U, \mathbb {Q} _l). \tag{6.2}
By the Weil conjectures [W1949], H^p(D_Q, \mathbb {Q} _l) is pure of weight p, so that E_2^{pq} is pure of weight p+2q.
As a quotient of a sub-object of E_2^{pq}, E_r^{pq} is also pure of weight p+2q.
By , d_r=0 for r \geq3, since the weights p+2q and p+2q-r+2 of E_r^{pq} and E_r^{p+q,q-r+1} (respectively) are different.
Thus E_3^{pq}=E_ \infty ^{pq}.
Up to renumbering, the weight filtration of H^ \bullet (U, \mathbb {Q} _l) is the abutment of :
312Equationhodge-theory-i-6.3hodge-theory-i-6.3.xml6.3hodge-theory-i-6 \operatorname {Gr} _n^W(H^k(U, \mathbb {Q} _l)) = E_3^{2k-n,n-k}. \tag{6.3} 315hodge-theory-i-7hodge-theory-i-7.xml7hodge-theory-i
In integer cohomology, for the usual topology, the Leray spectral sequence for j is of the form
314Equationhodge-theory-i-7.1hodge-theory-i-7.1.xml7.1hodge-theory-i-7 {'E_2}^{pq} = \bigoplus _{ \# Q=q} H^p(D_Q, \mathbb {Z} ) \Rightarrow H^{p+q}(U, \mathbb {Z} ). \tag{7.1}
Since each D_Q is a non-singular projective variety, {'E_2}^{pq} is endowed with a Hodge structure of weight p.
We set E_2^{pq}={'E_2}^{pq} \otimes \mathbb {Z} (-q) (a Hodge structure of weight p+2q).
As an abelian group, E_2^{pq}={'E_2}^{pq};
it is interesting to consider as a spectral sequence with initial page E_2^{pq}.
By , we should expect for d_2 \colon E_2^{pq} \to E_2^{p+2,q-1} to be a morphism of Hodge structures.
We prove this by thinking of d_2 as a Gysin morphism.
Then E_3^{pq} is endowed with a Hodge structure of weight p+2q.
By , we expect, modulo torsion, for the spectral sequence ([Trans.] The original refers to (6.4) instead of (6.2), but this seems to be a typo.) to degenerate at the E_3 page (i.e. E_3=E_ \infty), and for the vanishing of the d_r (for r \geq3) to be an application of .
This programme was successfully completed in [Hodge Theory II, 3.2].
There, we define the weight filtration of H^ \bullet (U, \mathbb {Q} ) as the abutment of , up to renumbering .
In fact, to endow the cohomology groups H^ \bullet with a mixed Hodge structure, the key point has always been, up until now, to find a spectral sequence E abutting to H^ \bullet such that the l-adic analogue of E_2^{pq} be conjecturally pure (of weight p+2q);
E_2^{pq} should then carry a natural Hodge structure (of weight p+2q), and the filtration W is then the abutment of E.
321hodge-theory-i-8hodge-theory-i-8.xml8hodge-theory-i
Let \operatorname {Spec} (V) be the spectrum of a Henselian discrete valuation ring (a Henselian trait) with field of fractions K, and residue field k that is of finite type over the prime field of characteristic p.
Let \overline {K} be an algebraic closure of K, and let H be a vector space of finite dimension over \mathbb {Q} _l (for l \neq p) on which \operatorname {Gal} ( \overline {K}/K) acts continuously.
By Grothendieck, we know ([ST1968, Appendix]) that a subgroup of finite index of the inertia group I acts unipotently.
By replacing V with a finite extension, we arrive to the case where the action of all of I is unipotent (the semi-stablecase);
it then factors as the largest pro-l-group I_l that is a quotient of I, and canonically isomorphic to \mathbb {Z} _l(1).
316Principlehodge-theory-i-principle-8.1hodge-theory-i-principle-8.1.xml8.1hodge-theory-i-8
In the semi-stable case, if the Galois module H "comes from algebraic geometry", then there exists a (unique) increasing filtration W of H (the "weight filtration") such that I acts trivially on \operatorname {Gr} _n^W(H), and such that \operatorname {Gr} _n^W(H), as a Galois module under \operatorname {Gal} ( \overline {k}/k) \simeq \operatorname {Gal} ( \overline {K}/K)/I, is pure of weight n.
We can compare this with and with the appendix of .
If we have a resolution of singularities, then we can sometimes give a conjectural definition of W whose validity follows from the Weil conjectures.
With the help of the resolution and of Weil, it is sometimes easy to show that, in any case, H splits into pure Galois modules (under \operatorname {Gal} ( \overline {k}/k)).
Suppose that H is semi-stable.
For T \in I_t, we define \log T by the finitesum - \sum _{n>0}( \Id-T )^n/n.
The map (T,x) \mapsto \log T(x) can be identified with a homomorphism
317Equationhodge-theory-i-8.2hodge-theory-i-8.2.xml8.2hodge-theory-i-8 M \colon \mathbb {Z} _l(1) \otimes H \to H. \tag{8.2}
Since \mathbb {Z} _l(1) is of weight -2, we necessarily have (cf. )
318Equationhodge-theory-i-8.3hodge-theory-i-8.3.xml8.3hodge-theory-i-8 M( \mathbb {Z} _l(1) \otimes W_n(H)) \subset W_{n-2}(H), \tag{8.3}
and M induces
319Equationhodge-theory-i-8.4hodge-theory-i-8.4.xml8.4hodge-theory-i-8 \operatorname {Gr} (M) \colon \mathbb {Z} _l(1) \otimes \operatorname {Gr} _n^W(H) \to \operatorname {Gr} _{n-2}^W(H). \tag{8.4} 320hodge-theory-i-8.5hodge-theory-i-8.5.xml8.5hodge-theory-i-8
If X is a non-singular projective variety over an algebraically closed field k_0, then we define
L \colon \mathbb {Z} _l(-1) \otimes H^ \bullet (X, \mathbb {Z} _l) \to H^ \bullet (X, \mathbb {Z} _l)
as being the cup product with the cohomology class of a hyperplane section.
We note that there is a formal analogy between L and M;
in the same way that M is defined by an action of \mathbb {Z} _l(1), we can think of L as being defined by an action of \mathbb {Z} _l(-1);
L increases the degree by 2, and \operatorname {Gr} M decreases it by 2.
354hodge-theory-i-9hodge-theory-i-9.xml9hodge-theory-i
Let D be the unit disc, D^*=D \setminus \{ 0 \}, and X
\begin {CD} X @>>> \mathbb {P} ^r( \mathbb {C} ) \times D \\ @VfVV @VV \mathrm {pr}_2V \\ D @= D \end {CD}
a family of projective varieties parameterised by D, with f proper, and f|D^* smooth.
Keeping the notation of , and recalling that, in the analogy between Henselian traits and small neighbourhoods of 0 in the complex line, we have the following dictionary (note that the spectrum of the ring of germs at 0 of holomorphic functions is a Henselian trait):
353Dictionaryhodge-theory-i-dictionary-9.1hodge-theory-i-dictionary-9.1.xml9.1hodge-theory-i-9
D
\operatorname {Spec} (V)
D^*
\operatorname {Spec} (K)
a universal covering \widetilde {D^*} of D^*
\operatorname {Spec} ( \overline {K})
the fundamental group \pi _1(D^*)
the inertia group I
(with \pi _1(D^*)= \mathbb {Z} \simeq \mathbb {Z} (1)_ \mathbb {Z})
(with I_l= \mathbb {Z} _l(1))
X
a projective scheme X over \operatorname {Spec} (V)
X^*=f^{-1}(D^*)
X_K
\widetilde {X}=X \times _D \widetilde {D^*}
X_{ \overline {K}}
the local system R^if_* \mathbb {Z} |D^*
the Galois module H^i(X_{ \overline {K}}, \mathbb {Z} _l)
H^i( \widetilde {X}, \mathbb {Z} )
H^i(X_{ \overline {K}}, \mathbb {Z} _l)
Note that \widetilde {X} is homotopically equivalent to each of the fibres X_t=f^{-1}(t) (for t \in D^*): H^i(X_{ \overline {K}}, \mathbb {Z} _l) is again analogous to H^i(X_t, \mathbb {Z} ), and the transformation of the monodromy T corresponds to the action of I.
Here, again, we know that a subgroup of finite index of \pi _1(D^*) acts unipotently on H^i( \widetilde {X}, \mathbb {Q} )=H^i(X_t, \mathbb {Q} ).
We place ourselves in the semi-stable case, where all of \pi _1(D^*) acts unipotently (this reduces to replacing D by a finite covering), and let T be the action of the canonical generator of \pi _1(D^*).
By and , we expect for H^i( \widetilde {X}, \mathbb {Q} ) \simeq H^i(X_t, \mathbb {Q} ) to be endowed with an increasing filtration W, for \operatorname {Gr} _n^W(H^i( \widetilde {X}, \mathbb {Q} )) to be endowed with a Hodge structure of weight n, for \log T(W_n) \subset W_{n-2}, and for \log T to induce a morphism of Hodge structures
M_n \colon \mathbb {Z} (-1) \otimes \operatorname {Gr} _n^W(H^i) \to \operatorname {Gr} _{n-2}^W(H^i).
We would further like for , and not just and , to have an analogue.
We have in fact managed to define, for each vector u of the tangent space to D at \{ 0 \}, a mixed Hodge structure \mathscr {H}_u on H^i( \widetilde {X}, \mathbb {Z} ).
The filtration W and the Hodge structures on the \operatorname {Gr} _n^W(H^i) are independent of u, and the dependence on u of \mathscr {H}_u can be expressed simply in terms of T.
Analogously to , we find that, for any u, \log T induces a homomorphism of mixed Hodge structures
M \colon \mathbb {Z} (1) \otimes H^i( \widetilde {X}, \mathbb {Z} ) \to H^i( \widetilde {X}, \mathbb {Z} ).
Finally, the analogy in is not misleading (but here, the fact that f|D^* is assumed to be proper and smooth is probably essential).
We can prove that
( \log T)^k \colon \operatorname {Gr} _{n+k}^W(H^n( \widetilde {X}, \mathbb {Q} )) \to \operatorname {Gr} _{n-k}^W(H^n( \widetilde {X}, \mathbb {Q} ))
is an isomorphism for all k (cf. [W1958, IV 6, Corollary to Theorem 5]).
This characterises the filtration W.
Until now, we have only had an analogue of the positivity theorem of Hodge (cf. [W1958, IV 7, Corollary to Theorem 7]) in very particular cases.
We hope that the mixed structures \mathscr {H}_u determine the asymptotic behaviour, for t \to0, of the family of pure structures H^i(X, \mathbb {Z} ) (for t \in D^*).
1696hodge-theory-iihodge-theory-ii.xmlHodge Theory II
P. Deligne.
"Théorie de Hodge II".
Pub. Math. de l'IHÉS 40 (1971) pp. 5–58.
publications.ias.edu/node/3611633hodge-theory-ii-introductionhodge-theory-ii-introduction.xmlIntroductionhodge-theory-iiWork presented as a doctoral thesis at l'Université d'Orsay.
By Hodge, the cohomology space \operatorname {H} ^n(X, \mathbb {C} ) of a compact Kähler variety X is endowed with a "Hodge structure" of weight n, i.e. a natural bigrading
\operatorname {H} ^n(X, \mathbb {C} ) = \bigoplus _{p+q=n} \operatorname {H} ^{p,q}
which satisfies \overline { \operatorname {H} ^{p,q}}= \operatorname {H} ^{q,p}.
We will show here that the complex cohomology of a non-singular, not necessarily compact, algebraic variety is endowed with a structure of a slightly more general type, which presents \operatorname {H} ^n(X, \mathbb {C} ) as a "successive extension" of Hodge structures of decreasing weights, contained between 2n and n, whose Hodge numbers h^{p,q}= \dim \operatorname {H} ^{p,q}, are zero for both p>n and q>n.
The reader will find an explanation in [Hodge Theory I] of the yoga that underlies this construction.
The proof, which is essentially algebraic, relies on one hand on Hodge theory, and on the other on Hironaka's resolution of singularities, which allows us, via a spectral sequence, to "express" the cohomology of a non-singular quasi-projective algebraic variety in terms of the cohomology of non-singular projective varieties.
contains, apart from reminders on filtrations gathered together for the ease of the reader, two key results:
, which will only be used via its corollary, [hodge-theory-ii-2.3.5 (?)], which gives the fundamental properties of "mixed Hodge structures".
The "two filtrations lemma", [hodge-theory-ii-1.3.16 (?)].
[hodge-theory-ii-2 (?)] recalls Hodge theory and introduces mixed Hodge structures.
The heart of this work is [hodge-theory-ii-3.2 (?)], which defines the mixed Hodge structure of \operatorname {H} ^n(X, \mathbb {C} ), and establishes some degenerations of spectral sequences.
[hodge-theory-ii-4 (?)] gives diverse applications, all following from [hodge-theory-ii-4.1.1 (?)] and the theory of the (K/k)-trace, for the resulting Hodge structures ([hodge-theory-ii-4.1.2 (?)]).
The principal ones are [hodge-theory-ii-4.2.6 (?)] and [hodge-theory-ii-4.4.15 (?)].
1638hodge-theory-ii-1hodge-theory-ii-1.xmlFiltrations1hodge-theory-ii1634hodge-theory-ii-1.1hodge-theory-ii-1.1.xmlFiltered objects1.1hodge-theory-ii-1579hodge-theory-ii-1.1.1hodge-theory-ii-1.1.1.xml1.1.1hodge-theory-ii-1.1
Let \mathscr {A} be an abelian category.
We will be considering \mathbb {Z}-filtrations, finite, in general, on the objects of \mathscr {A}:
580Definitionhodge-theory-ii-1.1.2hodge-theory-ii-1.1.2.xml1.1.2hodge-theory-ii-1.1
A decreasing (resp. increasing) filtration F of an object A of \mathscr {A} is a family (F^n(A))_{n \in \mathbb {Z} } (resp. (F_n(A))_{n \in \mathbb {Z} }) of sub-objects of A satisfying
\forall n,m \quad n \leqslant m \implies F^m(A) \subset F^n(A)
(resp. n \leqslant m \implies F_n(A) \subset F_m(A)).
A filtered object is an object endowed with a filtration.
When there is no chance of confusion, we often denote by the same letter filtrations on different objects of \mathscr {A}.
If F is a decreasing (resp. increasing) filtration on A, then we set F^ \infty (A)=0 and F^{- \infty }(A)=A (resp. F_{- \infty }(A)=0 and F_ \infty (A)=A).
The shifted filtrations of a decreasing filtration W are defined by
W[n]^p(A) = W^{n+p}(A)
for n \in \mathbb {Z}.
581hodge-theory-ii-1.1.3hodge-theory-ii-1.1.3.xml1.1.3hodge-theory-ii-1.1
If R is a decreasing (resp. increasing) filtration of A, then the F_n(A)=F^{-n}(A) (resp the F^n(A)=F_{-n}(A)) form an increasing (resp. decreasing) filtration of A.
This allows us in principal to consider only decreasing filtrations;
unless otherwise explicitly mentioned, when we say "filtration" we always mean "decreasing filtration".
582hodge-theory-ii-1.1.4hodge-theory-ii-1.1.4.xml1.1.4hodge-theory-ii-1.1
A filtration F of A is said to be finite if there exist n and m such that F^n(A)=A and F^m(A)=0.
583hodge-theory-ii-1.1.5hodge-theory-ii-1.1.5.xml1.1.5hodge-theory-ii-1.1
A morphism from a filtered object (A,F) to a filtered object (B,F) is a morphism f from A to B that satisfies f(F^n(A)) \subset F^n(B) for all n \in \mathbb {Z}.
Filtered objects (resp. finite filtered objects) of \mathscr {A} form an additive category in which inductive limits and finite projective limits exist (and thus kernels, cokernels, images, and coimages of a morphism).
A morphism f \colon (A,F) \to (B,F) is said to be strict, or strictly compatible with the filtrations, if the canonical arrows from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism of filtered objects (cf. ).
584hodge-theory-ii-1.1.6hodge-theory-ii-1.1.6.xml1.1.6hodge-theory-ii-1.1
Let (-)^ \circ be the contravariant identity functor from \mathscr {A} to the dual category \mathscr {A}^ \circ.
If (A,F) is a filtered object of \mathscr {A}, then the (A/F^n(A))^ \circ can be identified with sub-objects of A^ \circ.
The dual filtration on A^ \circ is defined by
F^n(A^ \circ ) = (A/F^{1-n})^ \circ .
The double dual of (A,F) can be identified with (A,F).
This construction identifies the dual of the category of filtered objects of \mathscr {A} with the category of filtered objects of \mathscr {A}^ \circ.
586hodge-theory-ii-1.1.7hodge-theory-ii-1.1.7.xml1.1.7hodge-theory-ii-1.1
If (A,F) is a filtered object of \mathscr {A}, then its associated graded is the object of \mathscr {A}^ \mathbb {Z} defined by
\operatorname {Gr} ^n(A) = F^n(A)/F^{n+1}(A).
The convention is justified by the simple formula
\operatorname {Gr} ^n(A^ \circ ) = \operatorname {Gr} ^{-n}(A)^ \circ
which follows from the self-dual diagram
585Equationhodge-theory-ii-1.1.7.1hodge-theory-ii-1.1.7.1.xml1.1.7.1hodge-theory-ii-1.1.7
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/F^n(A)
&& A/ F^{n+1}(A)
\ar [ll]
&& 0
\\ & A
\ar [ul] \ar [ur]
&& \operatorname {Gr} ^n(A)
\ar [ul] \ar [ur]
&& (1.1.7.1)
\\ F^{n+1}(A)
\ar [ur] \ar [rr]
&& F^n(A)
\ar [ul] \ar [ur]
&& 0
\ar [ul]
\end {tikzcd}
587hodge-theory-ii-1.1.8hodge-theory-ii-1.1.8.xml1.1.8hodge-theory-ii-1.1
Let (A,F) be a filtered object, and j \colon X \hookrightarrow A a sub-object of A.
The filtration induced by F (or simply the induced filtration) on X is the unique filtration on X such that j is strictly compatible with the filtrations;
we have
F^n(X) = j^{-1}(F^n(A)) = X \cap F^n(A).
Dually, the quotient filtration on A/X (the unique filtration such that p \colon A \to A/X is strictly compatible with the filtrations) is given by
F^n(A/X) = p(F^n(A)) \cong (X+F^n(A))/X \cong F^n(A)/(X \cap F^n(A)). 588Lemmahodge-theory-ii-1.1.9hodge-theory-ii-1.1.9.xml1.1.9hodge-theory-ii-1.1
If X and Y are sub-objects of A, with X \subset Y, then on Y/X \xrightarrow { \sim } \operatorname {Ker} (A/X \to A/Y) the quotient filtration of Y agrees with that induced by that of A/X.
In the diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
A/Y
&& A/X
\ar [ll]
\\ & A
\ar [ul] \ar [ur]
&& Y/X
\ar [ul]
\\ X
\ar [rr] \ar [ur]
&& Y
\ar [ul] \ar [uu] \ar [ur]
\end {tikzcd}
the arrows are strict.
589hodge-theory-ii-1.1.10hodge-theory-ii-1.1.10.xml1.1.10hodge-theory-ii-1.1
We call the filtration on Y/X the filtration induced by that of A (or simply the induced filtration).
By , its definition is self-dual.
In particular, if \Sigma \colon A \xrightarrow {f}B \xrightarrow {G}C is a **!!TO-DO: o-suite?!!** sequence, and if B is filtered, then \operatorname {H} ( \Sigma )= \operatorname {Ker} (g)/ \operatorname {Im} (f)= \operatorname {Ker} ( \operatorname {Coker} (f) \to \operatorname {Coim} (g)) is endowed with a canonical induced filtration.
The reader can show that:
593Propositionhodge-theory-ii-1.1.11hodge-theory-ii-1.1.11.xml1.1.11hodge-theory-ii-1.1
Let f \colon (A,F) \to (B,F) be a morphism of filtered objects with finite filtrations.
For f to be strict, it is necessary and sufficient that the sequence
0 \to \operatorname {Gr} ( \operatorname {Ker} (f)) \to \operatorname {Gr} (A) \to \operatorname {Gr} (B) \to \operatorname {Gr} ( \operatorname {Coker} (f)) \to 0
be exact.
Let \sigma \colon (A,F) \to (B,F) \to (C,F) be a **!!TO-DO: o-suite?!!** sequence of strict morphisms.
We then have
\operatorname {H} ( \operatorname {Gr} ( \Sigma )) \cong \operatorname {Gr} ( \operatorname {H} ( \Sigma ))
canonically.
In particular, if \Sigma is exact in \mathscr {A}, then \operatorname {Gr} ( \Sigma ) is exact in \mathscr {A}^ \mathbb {Z}.
In a category of modules, to say that a morphism f \colon (A,F) \to (B,F) is strict implies that every b \in B of filtration \geqslant n (i.e. b \in F^n(B)) that is in the image of A is already in the image of F^n(A):
f(F^n(A)) = f(A) \cap F^n(B). 594hodge-theory-ii-1.1.12hodge-theory-ii-1.1.12.xml1.1.12hodge-theory-ii-1.1
If \otimes \colon \mathscr {A}_1 \times \ldots \times \mathscr {A}_n \to \mathscr {B} is a multiadditive right-exact functor, and if A_i is an object of finite filtration of \mathscr {A}_i (for 1 \leqslant i \leqslant n), then we define a filtration on \bigotimes _{i=1}^n A_i by
F^k( \bigotimes _{i=1}^n A_i) = \sum _{ \sum k_i=k} \operatorname {Im} ( \bigotimes _{i=1}^n F^{k_i}(A_i) \to \bigotimes _{i=1}^n A_i)
(a sum of sub-objects).
Dually, if H is left-exact, then we set
F^k(H(A_i)) = \bigcap _{ \sum k_i=k} \operatorname {Ker} (H(A_i) \to H(A_i/F^{k_i}(A_i))).
If H is exact, then the two definitions are equivalent.
We extend these definitions to contravariant functors in certain variables by .
In particular, for the left-exact functor \operatorname {Hom}, we set
F^k( \operatorname {Hom} (A,B)) = \{ f \colon A \to B \mid f(F^n(A)) \subset F^{n+k}(B) \, \, \forall n \} .
We thus have
\operatorname {Hom} ((A,F),(B,F)) = F^0( \operatorname {Hom} (A,B)).
Under the above hypotheses, we have obvious morphisms
\begin {aligned} \bigotimes _{i=1}^n \operatorname {Gr} (A_i) & \to \operatorname {Gr} ( \bigotimes _{i=1}^n A_i) \\ \operatorname {Gr} H(A_i) & \to H( \operatorname {Gr} (A_i)). \end {aligned}
If H is exact, then these are isomorphisms and inverse to one another.
These constructions are compatible with composition of functors, in a sense whose details we leave to the reader.
1635hodge-theory-ii-1.2hodge-theory-ii-1.2.xmlOpposite filtrations1.2hodge-theory-ii-1657hodge-theory-ii-1.2.1hodge-theory-ii-1.2.1.xml1.2.1hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with filtrations F and G.
By definition, \operatorname {Gr} _F^n(A) is a quotient of a sub-object of A and, as such, is endowed with a filtration induced by G .
Passing to the associated graded defines a bigraded object ( \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A))_{n,m \in \mathbb {Z} }.
By a lemma of Zassenhaus, \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) and \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) are canonically isomorphic: if we define the induced filtrations as quotient filtrations of the induced filtrations on a sub-object, then we have
\begin {aligned} \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) \cong (F^m(A) \cap G^n(A)) &/ ((F^{m+1}(A) \cap G^n(A)) + (F^m(A) \cap G^{n+1}(A))) \\ &= \\ \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) \cong (G^n(A) \cap F^m(A)) &/ ((G^{n+1}(A) \cap F^m(A)) + (G^n(A) \cap F^{m+1}(A))) \end {aligned} 658hodge-theory-ii-1.2.2hodge-theory-ii-1.2.2.xml1.2.2hodge-theory-ii-1.2
Let H be a third filtration of A.
It induces a filtration on \operatorname {Gr} _F(A), and thus on \operatorname {Gr} _G \operatorname {Gr} _F(A).
It also induces a filtration on \operatorname {Gr} _F \operatorname {Gr} _G(A).
We note that these filtrations do not in general correspond to one another under the isomorphism .
In the expression \operatorname {Gr} _H \operatorname {Gr} _G \operatorname {Gr} _F(A), G and H thus play a symmetric role, but not F and G.
659Definitionhodge-theory-ii-1.2.3hodge-theory-ii-1.2.3.xml1.2.3hodge-theory-ii-1.2
Two finite filtrations F and \bar {F} on A are said to be n-opposite if \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q \neq n.
666hodge-theory-ii-1.2.4hodge-theory-ii-1.2.4.xml1.2.4hodge-theory-ii-1.2
If A^{p,q} is a bigraded object of \mathscr {A} such that
A^{p,q}=0 except for a finite number of pairs (p,q), and
A^{p,q}=0 for p+q \neq n
then we define two n-opposite filtrations of A= \sum _{p,q}A^{p,q} by setting
663Equationhodge-theory-ii-1.2.4.1hodge-theory-ii-1.2.4.1.xml1.2.4.1hodge-theory-ii-1.2.4 F^p(A) = \sum _{p' \geqslant p} A^{p',q'} \tag{1.2.4.1}
664Equationhodge-theory-ii-1.2.4.2hodge-theory-ii-1.2.4.2.xml1.2.4.2hodge-theory-ii-1.2.4 \bar {F}^q(A) = \sum _{q' \geqslant q} A^{p',q'}. \tag{1.2.4.2}
We have
665Equationhodge-theory-ii-1.2.4.3hodge-theory-ii-1.2.4.3.xml1.2.4.3hodge-theory-ii-1.2.4 \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A) = A^{p,q}. \tag{1.2.4.3}
Conversely:
667Propositionhodge-theory-ii-1.2.5hodge-theory-ii-1.2.5.xml1.2.5hodge-theory-ii-1.2
Let F and \bar {F} be finite filtrations on A.
For F and \bar {F} to be n-opposite, it is necessary and sufficient that, for all p,q,
[p+q=n+1] \implies [F^p(A) \oplus \bar {F}^q(A) \xrightarrow { \sim } A].
If F and \bar {F} are n-opposite, and if we set
\begin {cases} A^{p,q} = 0 & \text {for }p+q \neq n \\ A^{p,q} = F^p(A) \cap \bar {F}^q(A) & \text {for }p+q=n \end {cases}
then A is the direct sum of the A^{p,q}, and F and \bar {F} come from the bigrading A^{p,q} of A by the procedure of .
608Proofhodge-theory-ii-1.2.5
The condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n implies that F^p \cap \bar {F}^q=(F^{p+1} \cap \bar {F}^q)+(F^p \cap \bar {F}^{q+1}) for p+q>n.
By hypothesis, F^p \cap \bar {F}^q is zero for large enough p+q;
by decreasing induction, we thus deduce that the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n is equivalent to the condition F^p(A) \cap \bar {F}^q(A)=0 for p+q>n.
Dually (, , ), the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q<n is equivalent to the condition A=F^p(A)+ \bar {F}^q(A) for (1-p)+(1-q)>-n, i.e. for p+q \leqslant n+1, and the claim then follows.
If F and \bar {F} are n-opposite, then we can prove by decreasing induction on p that
604Equationhodge-theory-ii-1.2.5.1hodge-theory-ii-1.2.5.1.xml1.2.5.1hodge-theory-ii-1.2.5 \bigoplus _{p' \geqslant p} A^{p',q'} \xrightarrow { \sim } F^p(A). \tag{1.2.5.1}
For F^p(A)=0, the claim is evident.
The decomposition A=F^{p+1}(A) \oplus \bar {F}^{n-p}(A) induces on F^p(A) \supset F^{p+1}(A) a decomposition
F^p(A) = F^{p+1}(A) \oplus (F^p(A) \cap \bar {F}^{n-p}(A))
and we conclude by induction.
For p small enough, we have F^p(A)=A.
By , the A^{p,q} thus form a bigrading of A, and F satisfies .
The fact that \bar {F} satisfies then follows by symmetry.
668hodge-theory-ii-1.2.6hodge-theory-ii-1.2.6.xml1.2.6hodge-theory-ii-1.2
The constructions and establish equivalences of categories that are quasi-inverse to one another between objects of \mathscr {A} endowed with two finite n-opposite filtrations and bigraded objects of \mathscr {A} of the type considered in .
669Definitionhodge-theory-ii-1.2.7hodge-theory-ii-1.2.7.xml1.2.7hodge-theory-ii-1.2
Three finite filtrations W, F, and \bar {F} on A are said to be opposite if
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A) = 0
for p+q+n \neq0.
This condition is symmetric in F and \bar {F}.
It implies that F and \bar {F} induce on W^n(A)/W^{n+1}(A) two (-n)-opposite filtrations.
We set
A^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _F^{-p-q}(A)
whence decompositions ,
W^n(A)/W^{n+1}(A) = \bigoplus _{p+q=-n} A^{p,q} \tag{1.2.7.1}
which makes \operatorname {Gr} _W(A) into a bigraded object.
675Lemmahodge-theory-ii-1.2.8hodge-theory-ii-1.2.8.xml1.2.8hodge-theory-ii-1.2
Let W, F, and \bar {F} be three finite opposite filtrations, and \sigma a sequence (p_i,q_i)_{i \geqslant0 } pairs of integers satisfying
p_i \leqslant p_j and q_i \leqslant q_j for i \geqslant j, and
p_i+q_i=p_0+q_0-i+1 for i>0.
Set p=p_0, q=q_0, n=-p-q, and
A_ \sigma = \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap F^{p_i}(A)) \right ) \cap \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \right ).
Then the projection from W^n(A) to \operatorname {Gr} _W^n(A) induces an isomorphism
A_ \sigma \xrightarrow { \sim } A^{p,q} \subset \operatorname {Gr} _W^n(A).
674Proofhodge-theory-ii-1.2.8
We will prove by induction on k the following claim:
The projection from W^n(A)/W^{n+k} to \operatorname {Gr} _W^n(A) induces an isomorphism from
\begin {aligned} \Bigg [ & \left ( \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) + W^{n+k}(A) \right ) \\ \cap & \left ( \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) + W^{n+k}(A) \right ) \Bigg ] / W^{n+k}(A) \end {aligned} \tag{$*_k$}
to A^{p,q} \subset \operatorname {Gr} _W^n(A).
For k=1, this is exactly the definition of A^{p,q}.
By (i) of we have
673Equationhodge-theory-ii-1.2.8.1hodge-theory-ii-1.2.8.1.xml1.2.8.1hodge-theory-ii-1.2.8 F^{p_k}( \operatorname {Gr} _W^{n+k}(A)) \oplus \bar {F}^{q_k}( \operatorname {Gr} _W^{n+k}(A)) \xrightarrow { \sim } \operatorname {Gr} _W^{n+k}(A). \tag{1.2.8.1}
Set
\begin {aligned} B &= \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) \\ C &= \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \\ B' &= (W^{n+k}(A) \cap F^{p_k}(A)) + W^{n+k+1}(A) \\ C' &= (W^{n+k}(A) \cap \bar {F}^{q_k}(A)) + W^{n+k+1}(A) \\ D &= W^{n+k}(A) \\ E &= W^{n+k+1}(A). \end {aligned}
can then be written as
\begin {aligned} B'+C' &= D \\ B' \cap C' &= E. \end {aligned}
We also have, since p_k \leqslant p_i (for i \leqslant k),
B \cap D \subset F^{p_k}(A) \cap W^{n+k}(A) \subset B'
and, since q_k \leqslant q_i (for i \leqslant k),
C \cap D \subset \bar {F}^{q_k}(A) \cap W^{n+k}(A) \subset C'.
The claim (*_{k+1}) and then follows from (*_k) and the following lemma.
677Lemmahodge-theory-ii-1.2.9hodge-theory-ii-1.2.9.xml1.2.9hodge-theory-ii-1.2
Let B, C, B', C', D, and E be sub-objects of A.
Suppose that
\begin {gathered} B'+C' = D \qquad B' \cap C' = E \\ B \cap D \subset B' \qquad C \cap D \subset C'. \end {gathered}
Then
((B+B') \cap (C+C'))/E \xrightarrow { \sim } ((B+D) \cap (C+D))/D.
676Proofhodge-theory-ii-1.2.9
To prove surjectivity, we write
\begin {aligned} ((B+B') \cap (C+C'))+D &= (((B+B') \cap (C+C'))+B')+C' \\ &= ((B+B') \cap (C+C'+B'))+C' \\ &= (B+B'+C') \cap (C+C'+B') \\ &= (B+D) \cap (C+D). \end {aligned}
To prove injectivity, we write
(B+B') \cap (C+C') \cap D = ((B+B') \cap D) \cap ((C+C') \cap D).
Since B' \subset D, we have
\begin {aligned} (B+B') \cap D &= (B \cap D)+B' \\ &= B' \end {aligned}
and similarly
(C+C') \cap D = C'
and
\begin {aligned} (B+B') \cap (C+C') \cap D &= B' \cap C' \\ &= E. \end {aligned}
This finishes the proof of , noting that is equivalent to (*_k) for large k.
683Theoremhodge-theory-ii-1.2.10hodge-theory-ii-1.2.10.xml1.2.10hodge-theory-ii-1.2
Let \mathscr {A} be an abelian category, and provisionally denote by \mathscr {A}' the category of objects of \mathscr {A} endowed with three opposite filtrations W, F, and \bar {F}.
The morphisms in \mathscr {A}' are the morphisms of \mathscr {A} that are compatible with the three filtrations.
\mathscr {A}' is an abelian category.
The kernel (resp. cokernel) of an arrow f \colon A \to B in \mathscr {A}' is the kernel (resp. cokernel) of f in \mathscr {A}, endowed with the filtrations induced by those of A (resp. the quotients of those of B).
Every morphism f \colon A \to B in \mathscr {A}' is strictly compatible with the filtrations W, F, and \bar {F};
the morphism \operatorname {Gr} _W(f) is compatible with the bigradings of \operatorname {Gr} _W(A) and \operatorname {Gr} _W(B);
the morphisms \operatorname {Gr} _F(f) and \operatorname {Gr} _{ \bar {F}}(f) are strictly compatible with the filtration induced by W.
The "forget the filtrations" functors, \operatorname {Gr} _W, \operatorname {Gr} _F, and \operatorname {Gr} _{ \bar {F}}, and
\begin {gathered} \operatorname {Gr} _W \operatorname {Gr} _F \simeq \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _W \simeq \operatorname {Gr} _W \operatorname {Gr} _{ \bar {F}} \end {gathered}
from \mathscr {A}' to \mathscr {A} are exact.
Denote by \sigma _0(p,q) and \sigma _1(p,q) the sequences
\begin {aligned} \sigma _0(p,q) &= (p,q), (p,q), (p,q-1), (p,q-2), (p,q-3), \ldots \\ \sigma _0(p,q) &= (p,q), (p,q), (p-1,q), (p-2,q), (p-3,q), \ldots \end {aligned}
and, with the notation of , set
A_i^{p,q} = A_{ \sigma _i(p,q)} \qquad \text {for } i=0,1.
If f \colon A \to B is compatible with W, F, and \bar {F}, then we have
684Equationhodge-theory-ii-1.2.10.1hodge-theory-ii-1.2.10.1.xml1.2.10.1hodge-theory-ii-1.2 f(A_i^{p,q}) \subset B_i^{p,q} \qquad \text {for }i=0,1. \tag{1.2.10.1}
Claim (iii) then follows from the following lemma:
690Lemmahodge-theory-ii-1.2.11hodge-theory-ii-1.2.11.xml1.2.11hodge-theory-ii-1.2
The A_i^{p,q} give a bigrading of A.
We have
685Equationhodge-theory-ii-1.2.11.1hodge-theory-ii-1.2.11.1.xml1.2.11.1hodge-theory-ii-1.2.11 W^n(A) = \sum _{n+p+q \leqslant 0} A_i^{p,q} \qquad \text {for }i=0,1 \tag{1.2.11.1}
686Equationhodge-theory-ii-1.2.11.2hodge-theory-ii-1.2.11.2.xml1.2.11.2hodge-theory-ii-1.2.11 F^p(A) = \sum _{p' \geqslant p} A_0^{p',q'} \tag{1.2.11.2}
687Equationhodge-theory-ii-1.2.11.3hodge-theory-ii-1.2.11.3.xml1.2.11.3hodge-theory-ii-1.2.11 \bar {F}^q(A) = \sum _{q' \geqslant q} A_1^{p',q'}. \tag{1.2.11.3}
689Proofhodge-theory-ii-1.2.11
By symmetry, it suffices to prove the claims concerning i=0.
Set A_0= \bigoplus A_0^{p,q} and define filtrations W and F on A_0 by the equations of .
The canonical map i from A_0 to A is compatible with the filtrations W and F.
Furthermore, by , \operatorname {Gr} _W(i) is an isomorphism, and induces isomorphisms of graded objects
688Equationhodge-theory-ii-1.2.11.4hodge-theory-ii-1.2.11.4.xml1.2.11.4hodge-theory-ii-1.2.11 \sum _{p+q=n} A_0^{p,q} \xrightarrow { \sim } \operatorname {Gr} _W^{-n}(A) = \sum _{p+q=n} A^{p,q}. \tag{1.2.11.4}
The morphism i is thus an isomorphism, and the A_0^{p,q} give a bigrading of A.
then says that \operatorname {Gr} _W(i) is an isomorphism.
By , \operatorname {Gr} _F \operatorname {Gr} _W(i) is an isomorphism, and thus so too are \operatorname {Gr} _W \operatorname {Gr} _F(i) and \operatorname {Gr} _F(i).
Equation then follows.
691hodge-theory-ii-1.2.12hodge-theory-ii-1.2.12.xml1.2.12hodge-theory-ii-1.2
We now prove .
Let f \colon A \to B in \mathscr {A}' and endow K= \operatorname {Ker} (f) with the filtrations induced by those of A.
By , \operatorname {Gr} _W(K) \hookrightarrow \operatorname {Gr} _W(A);
furthermore, the filtration F (resp. \bar {F}) on K induces on \operatorname {Gr} _W(K) the inverse image filtration of the filtration F on \operatorname {Gr} _W(A).
The sub-object \operatorname {Gr} _W(K) of \operatorname {Gr} _W(A) is then compatible with the bigrading of \operatorname {Gr} _W(A):
\operatorname {Gr} _W(K) = \bigoplus _{p,q}( \operatorname {Gr} _W(K) \cap A^{p,q}).
We thus deduce that
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(K) \hookrightarrow \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A);
the filtrations of W, F, and \bar {F} on K are thus opposite, and K is a kernel of f in \mathscr {A}'.
This, combined with the dual result, proves (ii).
If f is an arrow of \mathscr {A}', then the canonical morphism from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism in \mathscr {A};
by (iii), it is also an isomorphism in \mathscr {A}', which is thus abelian.
The "forget the filtrations" functor is exact by (ii).
The exactness of the other functors in (iv) follows immediately from (ii), (iii), and (i) or (ii) of .
692hodge-theory-ii-1.2.13hodge-theory-ii-1.2.13.xml1.2.13hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with a finite increasing filtration W_ \bullet, and two finite decreasing filtrations F and \bar {F}.
The construction associates to W_ \bullet a decreasing filtration W^ \bullet.
We say that the filtrations W_ \bullet, F, and \bar {F} are opposite if the filtrations W^ \bullet, F, and \bar {F} are, i.e. if, for all n, the filtrations induced by F and \bar {F} on
\operatorname {Gr} _n^W(A) = W_n(A)/W_{n-1}(A)
are n-opposite.
translates trivially to this variation.
1636hodge-theory-ii-1.3hodge-theory-ii-1.3.xmlThe two filtrations lemma1.3hodge-theory-ii-1823hodge-theory-ii-1.3.1hodge-theory-ii-1.3.1.xml1.3.1hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with a filtration F.
The filtration is said to be biregular if it induces a finite filtration on each component of K.
We recall the definition of the terms E_r^{pq}(K,F), or simply E_r^{pq}, of the spectral sequence defined by F.
We set
Z_r^{pq} = \operatorname {Ker} (d \colon F^p(K^{p+q}) \to F^{p+q+1}/F^{p+r}(K^{p+q+1}))
and dually we define B_r^{pq} by the formula
K^{p+q}/B_r^{pq} = \operatorname {Coker} (d \colon F^{p-r+1}(K^{p+q+1}) \to K^{p+q}/F^{p+1}(K^{p+q})).
These formulas still make sense for r= \infty.
We note that the use here of the notation B_r^{pq} is different to that of Godement [G1958].
We have, by definition:
395Equationhodge-theory-ii-1.3.1.1hodge-theory-ii-1.3.1.1.xml1.3.1.1hodge-theory-ii-1.3.1 E_r^{pq} = \operatorname {Im} (Z_r^{pq} \to K^{p+q}/B_r^{pq}) \tag{1.3.1.1}
396Equationhodge-theory-ii-1.3.1.2hodge-theory-ii-1.3.1.2.xml1.3.1.2hodge-theory-ii-1.3.1 = Z_r^{pq}/(B_r^{pq} \cap Z_r^{pq}) \tag{1.3.1.2}
397Equationhodge-theory-ii-1.3.1.3hodge-theory-ii-1.3.1.3.xml1.3.1.3hodge-theory-ii-1.3.1 = \operatorname {Ker} (K^{p+q}/B_r^{pq} \to K^{p+q}/(Z_r^{pq}+B_r^{pq})). \tag{1.3.1.3}
We can thus write
398Equationhodge-theory-ii-1.3.1.4hodge-theory-ii-1.3.1.4.xml1.3.1.4hodge-theory-ii-1.3.1 \begin {aligned} B_r^{p \bullet } \cap Z_r^{p \bullet } & \coloneqq (dF^{p-r+1}+F^{p+1}) \cap (d^{-1}F^{p+r} \cap F^p) \\ &= (dF^{p-r+1} \cap F^p) + (F^{p+1} \cap d^{-1}F^{p+r}) \end {aligned} \tag{1.3.1.4}
since dF^{p-r+1} \subset d^{-1}F^{p+r} and F^{p+1} \subset F^p.
For r< \infty, the E_r form a complex graded by the degree p-r(p+q), and E_{r+1} can be expressed as the cohomology of this complex:
399Equationhodge-theory-ii-1.3.1.5hodge-theory-ii-1.3.1.5.xml1.3.1.5hodge-theory-ii-1.3.1 E_{r+1}^{pq} = \operatorname {H} (E_r^{p-r,q+r-1} \xrightarrow {d_r} E_r^{pq} \xrightarrow {d_r} E_r^{p+r,q-r+1}). \tag{1.3.1.5}
For r=0, we have
400Equationhodge-theory-ii-1.3.1.6hodge-theory-ii-1.3.1.6.xml1.3.1.6hodge-theory-ii-1.3.1 E_0^{ \bullet \bullet } = \operatorname {Gr} _F^ \bullet (K^ \bullet ). \tag{1.3.1.6} 828Propositionhodge-theory-ii-1.3.2hodge-theory-ii-1.3.2.xml1.3.2hodge-theory-ii-1.3
Let K be a complex endowed with a biregular filtration F.
The following conditions are equivalent:
The spectral sequence defined by F degenerates (E_1=E_ \infty).
The morphisms d \colon K^i \to K^{i+1} are strictly compatible with the filtrations.
827Proofhodge-theory-ii-1.3.2
We will prove this in the case where \mathscr {A} is a category of modules.
For fixed p and q, the hypothesis that the arrows d_r with domains E_r^{pq} be zero for r \geqslant1 implies that, if x \in F^p(K^{p+q}) satisfies dx \in F^{p+1}(K^{p+q+1}), then there exists y \in K^{p+q} such that dy=0 and such that x and y have the same image in E_1^{pq}.
Modifying y by a boundary, and setting z=x-y, we then have
\forall x \in F^p(K^{p+q}) \left [ dx \in F^{p+1}(K^{p+q+1}) \implies \exists z \text { s.t. } z \in F^{p+1}(K^{p+q}) \text { and } dz=dx \right ]
or, in other words,
F^{p+1}(K^{p+q+1}) \cap dF^p(K^{p+q}) = dF^{p+1}(K^{p+q}). \tag{1}
If this condition is satisfied for arbitrary p and q, then by induction on r we have
F^{p+r} \cap dF^p = dF^{p+r}
which, for large p+r, can be written as
F^p \cap dK = dF^p. \tag{2}
Claim (2) trivially implies (1), and is equivalent to (ii), which proves the proposition.
831hodge-theory-ii-1.3.3hodge-theory-ii-1.3.3.xml1.3.3hodge-theory-ii-1.3
If (K,F) is a filtered complex, we denote by \operatorname {Dec} (K) the complex K endowed with the shifted filtration
\operatorname {Dec} (F)^p K^n = Z_1^{p+n,-p}.
This filtration is compatible with the differentials:
\begin {aligned} dZ_1^{p+n,-p} & \subset F^{p+n+1}(K^{n+1}) \cap \operatorname {Ker} (d) \\ & \subset Z_ \infty ^{p+n+1,-p} \\ & \subset Z_1^{p+n+1,-p}. \end {aligned}
Since
829Equationhodge-theory-ii-1.3.3.1hodge-theory-ii-1.3.3.1.xml1.3.3.1hodge-theory-ii-1.3.3 \begin {aligned} Z_1^{p+1+n,-p-1} & \subset F^{p+1+n}(K^n) \\ & \subset B_1^{p+n,-p} \\ & \subset Z_1^{p+n,-p} \end {aligned} \tag{1.3.3.1}
the evident arrow from Z_1^{p+n,-p}/Z_1^{p+1+n,-p-1} to Z_1^{p+n,-p}/B_1^{p+n,-p} is a morphism
830Equationhodge-theory-ii-1.3.3.2hodge-theory-ii-1.3.3.2.xml1.3.3.2hodge-theory-ii-1.3.3 u \colon E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,-p}(K). \tag{1.3.3.2} 838Propositionhodge-theory-ii-1.3.4hodge-theory-ii-1.3.4.xml1.3.4hodge-theory-ii-1.3
The morphisms in form a morphism of graded complexes from E_0( \operatorname {Dec} (K)) to E_1(K).
This morphism induces an isomorphism on cohomology.
This morphism induces step-by-step (via ) isomorphisms of graded complexes E_r( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}(K) (for r \geqslant1).
837Proofhodge-theory-ii-1.3.4
Let F' be the filtration on K defined by
{F'}^p(K^n) = \operatorname {Dec} (F)^{p-n}(K^n) = Z_1^{p,n-p}.
We trivially have isomorphisms
836Equationhodge-theory-ii-1.3.4.1hodge-theory-ii-1.3.4.1.xml1.3.4.1hodge-theory-ii-1.3.4 E_r^{p,n-p}( \operatorname {Dec} (K)) = E_{r+1}^{p+n,-p}(K,F') \tag{1.3.4.1}
that are compatible with the d_r and with .
The map u comes from and from the identity map
(K,F') \to (K,F).
This proves (i), and it remains to show that, for r \geqslant2,
E_r^{pq}(K,F') \xrightarrow { \sim } E_r^{pq}(K,F).
We have
\begin {aligned} Z_r^{pq}(K,F') &= Z_r^{pq}(K,F) \qquad \text {for }r \geqslant1 \\ Z_r^{pq}(K,F') \cap B_r^{pq}(K,F') &= Z_r^{pq}(K,F) \cap B_r^{pq}(K,F) \qquad \text {for }r \geqslant2 \end {aligned}
and we can then apply .
839hodge-theory-ii-1.3.5hodge-theory-ii-1.3.5.xml1.3.5hodge-theory-ii-1.3
The construction in is not self-dual.
The dual construction consists of defining
\operatorname {Dec} ^ \bullet (F)^pK^n = B_1^{p+n-1,-p+1}.
We then have morphisms
E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,p}(K) \to E_0^{p,n-p}( \operatorname {Dec} ^ \bullet (K))
and, for r \geqslant1, isomorphisms
E_r^{p,n-p}( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}^{p+n,p}(K) \xrightarrow { \sim } E_{r}^{p,n-p}( \operatorname {Dec} ^ \bullet (K)).
Recall that a morphism of complexes is said to be a quasi-isomorphism if it induces an isomorphism on cohomology.
843Definitionhodge-theory-ii-1.3.6hodge-theory-ii-1.3.6.xml1.3.6hodge-theory-ii-1.3
A morphism f \colon (K,F) \to (K',F') of filtered complexes with biregular filtrations is a filtered quasi-isomorphism if \operatorname {Gr} _F(f) is a quasi-isomorphism, i.e. if the E_1^{pq}(f) are isomorphisms.
A morphism f \colon (K,F,W) \to (K,F',W') of biregular bifiltered complexes is a bifiltered quasi-isomorphism if \operatorname {Gr} _F \operatorname {Gr} _W(f) is a quasi-isomorphism.
844hodge-theory-ii-1.3.7hodge-theory-ii-1.3.7.xml1.3.7hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with two filtrations F and W.
Let E_r^{pq} be the spectral sequence defined by W.
The filtration F induces on E_r^{pq} various filtrations, which we will compare.
845hodge-theory-ii-1.3.8hodge-theory-ii-1.3.8.xml1.3.8hodge-theory-ii-1.3 identifies E_r^{pq} with a quotient of a sub-object of K^{p+q}.
The E_r^{pq} term is thusly given by endowing with a filtration F_d induced by F, called the first direct filtration.
846hodge-theory-ii-1.3.9hodge-theory-ii-1.3.9.xml1.3.9hodge-theory-ii-1.3
Dually, identifies E_r^{pq} with a sub-object of a quotient of K^{p+q}, whence a new filtration F_{d^*} induced by F, called the second direct filtration.
847Lemmahodge-theory-ii-1.3.10hodge-theory-ii-1.3.10.xml1.3.10hodge-theory-ii-1.3
On E_0 and E_1, we have F_d=F_{d^*}.
651Proofhodge-theory-ii-1.3.10
For r=0,1, we have B_r^{pq} \subset Z_r^{pq}, and we apply .
851hodge-theory-ii-1.3.11hodge-theory-ii-1.3.11.xml1.3.11hodge-theory-ii-1.3 identifies E_{r+1}^{pq} with a quotient of a sub-object of E_r^{pq}.
We define the recurrent filtration F_r on the E_r^{pq} by the conditions
On E_0^{pq}, F_r=F_d=F_{d^*}.
On E_{r+1}^{pq}, the recurrent filtration is that induced by the recurrent filtration of E_r^{pq}.
852hodge-theory-ii-1.3.12hodge-theory-ii-1.3.12.xml1.3.12hodge-theory-ii-1.3
Definitions and still make sense for r= \infty.
If the filtration on K is biregular, then the direct filtrations on E_ \infty ^{pq} coincide with those on E_r^{pq}=E_ \infty ^{pq} for large enough r, and we define the recurrent filtration on E_ \infty ^{pq} as agreeing with that on E_r^{pq} for large enough r.
The filtrations F and W each induce a filtration on H^ \bullet (K), and E_ \infty ^{ \bullet \bullet }= \operatorname {Gr} _W^ \bullet ( \operatorname {H} ^ \bullet (K)).
The filtration F on \operatorname {H} ^ \bullet (K) then induces on E_ \infty ^{pq} a new filtration.
859Propositionhodge-theory-ii-1.3.13hodge-theory-ii-1.3.13.xml1.3.13hodge-theory-ii-1.3
For the first direct filtration, the morphisms d_r are compatible with the filtrations.
If E_{r+1}^{pq} is considered as a quotient of a sub-object of E_r^{pq}, then the first direct filtration on E_{r+1}^{pq} is finer than the filtration F' induced by the first direct filtration on E_r^{pq}Y we have F_d(E_{r+1}^{pq}) \subset F'(E_{r+1}^{pq}).
Dually, the morphisms d_r are compatible with the second direct filtration, and the second direct filtration on E_{r+1}^{pq} is less fine than the filtration induced by that of E_r^{pq}.
F_d(E_r^{pq}) \subset F_r(E_r^{pq}) \subset F_{d^*}(E_r^{pq}).
On E_ \infty ^{pq}, the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K) is finer than the first direct filtration and less fine than the second.
858Proofhodge-theory-ii-1.3.13
Claim (i) is evident, (ii) is its dual, and (iii) follows by induction.
The first claim of (iv) is easy to verify, and the second is its dual.
860hodge-theory-ii-1.3.14hodge-theory-ii-1.3.14.xml1.3.14hodge-theory-ii-1.3
We denote by \operatorname {Dec} (K) (resp. \operatorname {Dec} ^ \bullet (K)) the complex K endowed with the filtrations \operatorname {Dec} (W) and F (resp. \operatorname {Dec} ^ \bullet (W) and F).
It is clear by that the isomorphism in sends the first direct filtration on E_r( \operatorname {Dec} (K)) to the second direct filtration on E_{r+1}(K) (for r \geqslant1).
The dual isomorphism sends the second direct filtration on E_r( \operatorname {Dec} ^ \bullet (K)) to the second direct filtration on E_{r+1}(K).
865Lemmahodge-theory-ii-1.3.15hodge-theory-ii-1.3.15.xml1.3.15hodge-theory-ii-1.3
If the filtration F is biregular, and if, on the \operatorname {Gr} _W^p(K), the morphisms d are strictly compatible with the filtration induced by F, then
The morphism of graded complexes filtered by F
u \colon \operatorname {Gr} _{ \operatorname {Dec} (W)}(K) \to E_1(K,W)
is a filtered quasi-isomorphism.
Dually, the morphism in
u \colon E_1(K,W) \to \operatorname {Gr} _{ \operatorname {Dec} ^ \bullet (W)}(K)
is a filtered quasi-isomorphism.
864Proofhodge-theory-ii-1.3.15
It suffices, by duality, to prove (i).
By and , the complex E_1(K,W) filtered by F is a quotient of the filtered complex \operatorname {Gr} _{ \operatorname {Dec} (W)}(K).
Let U be the filtered complex given by the kernel, which is acyclic by (ii) of .
The long exact sequence in cohomology associated to the exact sequence of complexes
0 \to \operatorname {Gr} _F(U) \to \operatorname {Gr} _F( \operatorname {Gr} _{ \operatorname {Dec} (W)}(K)) \to \operatorname {Gr} _F(E_1(K,W)) \to 0
shows that u is a filtered quasi-isomorphism if and only if \operatorname {Gr} _F(U) is an acyclic complex.
By , and since U is acyclic, this reduces to asking that the differentials of U be strictly compatible with the filtration F.
From we obtain that U is the sum over p of the complexes
(U^p)^n = B_1^{p+n,-p}/Z_1^{p+1+n,-p-1}
endowed with the filtration induced by F.
Each differential d of each of the complexes U^p fits into a commutative diagram of filtered objects of the following type, where, for simplicity, we have omitted the total or complementary degree:
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
B^p/Z^{p+1}
\ar [rrr,"d"]
\ar [dr]
&&& B^{p+1}/Z^{p+1}
\ar [dd]
\\ & \operatorname {Coim} (d)
\ar [r,"(5)"]
& \operatorname {Im} (d)
\ar [ur] \ar [dr]
\\ W^{p+1}/Z^{p+1}
\ar [uu]
\ar [ur]
\ar [rrr,near start,swap,"(3)"]
\ar [white,urr,near end,swap," \color {black}(4)"]
&&& B^{p+1}/W^{p+2}
\ar [d]
\\ W^{p+1}/W^{p+2}
\ar [u]
\ar [rrr,near start,swap,"(1)"]
\ar [white,urrr," \color {black}(2)"description]
&&& W^{p+1}/W^{p+2}
\end {tikzcd}
By hypothesis, the morphism (1) is strict.
Since the square (2) is exactly the canonical decomposition of (1), the arrow (3) is a filtered isomorphism.
The arrows of the trapezium (4) are isomorphisms;
they are thus filtered isomorphisms, since (3) is a filtered isomorphism.
The fact that (5) is a filtered isomorphism implies that d is strict.
This proves the lemma.
867Theoremhodge-theory-ii-1.3.16hodge-theory-ii-1.3.16.xml1.3.16hodge-theory-ii-1.3
Let K be a complex endowed with two filtrations, W and F, with the filtration F biregular.
Let r_0 \geqslant0 be an integer, and suppose that, for 0 \leqslant r<r_0, the differentials of the graded complex E_r(K,W) are strictly compatible with the filtration F.
Then, for r \leqslant r_0+1, F_d=F_r=F_{d^*} on E_r^{pq}.
866Proofhodge-theory-ii-1.3.16
We will prove the theorem by induction on r_0.
For r_0=0, the hypothesis is empty, and we apply and (iii) of .
For r_0 \geqslant1, by the inductive hypothesis, we have F_d=F_r=F_{d^*} on E_r^{pq} for r \leqslant r_0.
By , the morphism u \colon E_0( \operatorname {Dec} (K)) \to E_1(K) is a filtered quasi-isomorphism.
It thus induces a filtered isomorphism from \operatorname {H} ^ \bullet ( \operatorname {Dec} (K)) to \operatorname {H} ^ \bullet (E_1(K)):
u \colon (E_1( \operatorname {Dec} (K)),F_r) \xrightarrow { \sim } (E_2(K),F_r).
Step-by-step, we thus deduce that the canonical isomorphism from E_s( \operatorname {Dec} (K)) to E_{s+1} (for s \geqslant1) is a filtered isomorphism for the recurrent filtration.
On E_1( \operatorname {Dec} (K)), F_r=F_d (), and we already know () that u' is a filtered isomorphism
u' \colon (E_1( \operatorname {Dec} (K)),F_d) \xrightarrow { \sim } (E_2(K),F_d).
On E_2(K), we thus have F_d=F_r.
This, combined with the dual result, proves the theorem for r_0=1.
Suppose that r_0 \geqslant2.
Then the arrows d_1 of E_1(K) are strictly compatible with the filtrations, and thus so too are the arrows d_0 of E_0( \operatorname {Dec} (K)) (indeed, u induces an isomorphism of spectral sequences, and we apply the criterion from ).
For 0<s<r_0-1, the isomorphism (E_s( \operatorname {Dec} (K)),F_r) \cong (E_{s+1}(K),F_r) shows that the d_s are strictly compatible with the recurrent filtrations.
By the induction hypothesis, we thus have F_d=F_r on E_s( \operatorname {Dec} (K)) for s \leqslant s_0.
The isomorphism (E_s( \operatorname {Dec} (K)),F_d) \cong (E_{s+1}(K),F_d) () then shows that F_d=F_r on E_r(K) for r \leqslant r_0+1.
This, combined with the dual result, proves the theorem.
869Corollaryhodge-theory-ii-1.3.17hodge-theory-ii-1.3.17.xml1.3.17hodge-theory-ii-1.3
Under the general hypotheses of , suppose that, for all r, the differentials d_r are strictly compatible with the recurrent filtrations on the E_r.
Then, on E_ \infty, the filtrations F_d, F_r, and F_{d^*} agree, and coincide with the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K).
868Proofhodge-theory-ii-1.3.17
This follows immediately from and (iv) of .
1637hodge-theory-ii-1.4hodge-theory-ii-1.4.xmlHypercohomology of filtered complexes1.4hodge-theory-ii-1
In this section, we recall some standard constructions in hypercohomology.
We do not use the language of derived categories, which would be more natural here.
Throughout this entire section, by "complex" we mean "bounded-below complex".992hodge-theory-ii-1.4.1hodge-theory-ii-1.4.1.xml1.4.1hodge-theory-ii-1.4
Let T be a left-exact functor from an abelian category \mathscr {A} to an abelian category \mathscr {B}.
Suppose that every object of \mathscr {A} injects into an injective object;
the derived functors \mathrm {R} ^iT \colon \mathscr {A} \to \mathscr {B} are then defined.
An object A of \mathscr {A} is said to be *acyclic* for T if \mathrm {R} ^iT(A)=0 for i>0.
994hodge-theory-ii-1.4.2hodge-theory-ii-1.4.2.xml1.4.2hodge-theory-ii-1.4
Let (A,F) be a filtered object with finite filtration, and TF the filtration of TA by its sub-objects TF^p(A) (these are sub-objects since T is left exact).
If \operatorname {Gr} _F(A) is T-acyclic, then the F^p(A) are T-acyclic as successive extensions of T-acyclic objects.
The image under T of the sequence
0 \to F^{p+1}(A) \to F^p(A) \to \operatorname {Gr} ^p(A) \to 0
is thus exact, and
993Equationhodge-theory-ii-1.4.2.1hodge-theory-ii-1.4.2.1.xml1.4.2.1hodge-theory-ii-1.4.2 \operatorname {Gr} _{FT}TA \xrightarrow { \sim } T \operatorname {Gr} _FA. \tag{1.4.2.1} 995hodge-theory-ii-1.4.3hodge-theory-ii-1.4.3.xml1.4.3hodge-theory-ii-1.4
Let A be an object endowed with finite filtrations F and W such that \operatorname {Gr} _F \operatorname {Gr} _W A are T-acyclic.
The objects \operatorname {Gr} _FA and \operatorname {Gr} _WA are then T-acyclic, as well as the F^q(A) \cap W^p(A).
The sequences
0 \to T(F^q \cap W^{p+1}) \to T(F^q \cap W^p) \to T((F^q \cap W^p)/(F^q \cap W^{p+1})) \to 0
are thus exact, and T(F^q( \operatorname {Gr} _W^p(A))) is the image in T( \operatorname {Gr} _W^p(A)) of T(F^p \cap W^q).
The diagram
\begin {CD} T(F^q \cap W^p) @>>> T(F^q \operatorname {Gr} _W^pA) @>>> T \operatorname {Gr} _W^pA \\ @V{ \cong }VV @. @VV{ \cong }V \\ TF^q \cap TW^p @= TF^q \cap TW^p @>>> \operatorname {Gr} _{TW}^pTA \end {CD}
then shows that the isomorphism in relative to W sends the filtration \operatorname {Gr} _{TW}(TF) to the filtration T( \operatorname {Gr} _W(F)).
999hodge-theory-ii-1.4.4hodge-theory-ii-1.4.4.xml1.4.4hodge-theory-ii-1.4
Let K be a complex of objects of \mathscr {A}.
The hypercohomology objects \mathrm {R} ^iT(K) are calculated as follows:
We choose a quasi-isomorphism i \colon K \to K such that the components of K' are acyclic for T.
For example, we can take K' to be the simple complex associated to an injective Cartan–Eilenberg resolution of K.
We set
\mathrm {R} ^iT(K) = \operatorname {H} ^i(T(K')).
We can show that \mathrm {R} ^iT(K) does not depend on the choice of K', but depends functorially on K, and that a quasi-isomorphism f \colon K_1 \to K_2 induces *isomorphisms*
\mathrm {R} ^iT(f) \colon \mathrm {R} ^iT(K_1) \to \mathrm {R} ^iT(K_2). 1000hodge-theory-ii-1.4.5hodge-theory-ii-1.4.5.xml1.4.5hodge-theory-ii-1.4
Let F be a biregular filtration of K.
A {#T}-acyclic filtered resolution of K is a filtered quasi-isomorphism i \colon K \to K' from K to a filtered biregular complex such that the \operatorname {Gr} ^p({K'}^n) are acyclic for T.
If K' is such a resolution, then the {K'}^n are acyclic for T, and the filtered complex (cf. ) T(K') defines a spectral sequence
E_1^{pq} = \mathrm {R} ^{p+q}T( \operatorname {Gr} ^p(K)) \Rightarrow \mathrm {R} ^{p+q}T(K).
This is independent of the choice of K'.
We call this the hypercohomology spectral sequence of the filtered complex K.
It depends functorially on K, and a filtered quasi-isomorphism induces an isomorphism of spectral sequences.
The differentials d_1 of this spectral sequence are the connection morphisms defined by the short exact sequences
0 \to \operatorname {Gr} ^{p+1}K \to F^pK/F^{p+2}K \to \operatorname {Gr} ^pK \to 0. 1001hodge-theory-ii-1.4.6hodge-theory-ii-1.4.6.xml1.4.6hodge-theory-ii-1.4
Let K be a complex.
We denote by \tau _{ \leqslant p}(K) the following subcomplex:
\tau _{ \leqslant p}(K)^n = \begin {cases} K^n & \text {for }n<p \\ \operatorname {Ker} (d) & \text {for }n=p \\ 0 & \text {for }n>p. \end {cases}
The filtration, said to be canonical, of K by the \tau _{ \leqslant p}(K) is induced by shifting the trivial filtration G for which G^0(K)=K and G^1(K)=0.
We have, for the canonical filtration,
\begin {aligned} E_1^{pq} = 0 & \qquad \text {if }p+q \neq-p \\ \\ H^{-p} & \qquad \text {if }p+q=-p. \end {aligned}
A quasi-isomorphism f \colon K \to K' is automatically a filtered quasi-isomorphism for the canonical filtrations.
1002hodge-theory-ii-1.4.7hodge-theory-ii-1.4.7.xml1.4.7hodge-theory-ii-1.4
The subcomplexes \sigma _{ \geqslant p}(K) of K
\sigma _{ \geqslant p}(K)^n = \begin {cases} 0 & \text {if }n<p \\ K^n & \text {if }n \geqslant p \end {cases}
define a biregular filtration, called the stupid filtration of K.
The hypercohomology spectral sequences attached to the stupid or canonical filtrations of K are the two hypercohomology spectral sequences of K.
1003Examplehodge-theory-ii-1.4.8hodge-theory-ii-1.4.8.xml1.4.8hodge-theory-ii-1.4
Let f \colon X \to Y be a continuous map between topological spaces, and let \mathscr {F} be an abelian sheaf on X.
Let \mathscr {F}^ \bullet be a resolution of \mathscr {F} by f_*-acyclic sheaves.
We have \mathrm {R} ^i f_* \mathscr {F} \simeq \operatorname { \mathscr {H}} ^i(f_* \mathscr {F}^ \bullet ).
We take the functor T to be the functor \Gamma (Y,-).
The hypercohomology spectral sequence of the complex f_* \mathscr {F}^ \bullet endowed with its canonical filtration
E_1^{pq} = \operatorname {H} ^{2p+q}(Y, \mathrm {R} ^{-p}f_* \mathscr {F}) \Rightarrow \operatorname {H} ^{p+q}(X, \mathscr {F})
is exactly, up to the renumbering E_r^{pq} \mapsto E_{r+1}^{2p+q,-p}, the Leray spectral sequence for f and \mathscr {F}.
1010hodge-theory-ii-1.4.9hodge-theory-ii-1.4.9.xml1.4.9hodge-theory-ii-1.4
Let (K,W,F) be a biregular bifiltered complex.
To this complex, we associate:
A spectral sequence
{}_W E_1^{p,n-p} = \operatorname {H} ^n( \operatorname {Gr} _W^p(K)) \Rightarrow \operatorname {H} ^n(K)
with differentials {}_W d_1 being the connecting morphisms induced by the short exact sequences
0 \to \operatorname {Gr} _W^{p+1}(K) \to W^p(K)/W^{p+2}(K) \to \operatorname {Gr} _W^p(K) \to 0;
An analogous spectral sequence for the filtration F;
Exact squares
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& 0 \dar
& 0 \dar
& 0 \dar
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^{q+1}K \rar \dar
& F^p/F^{p+2}( \operatorname {Gr} _W^{q+1}K) \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1}K \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1}(W^q/W^{q+2}(K)) \rar \dar
& F^p/F^{p+2}(W^q/W^{q+2}(K)) \rar \dar
& \operatorname {Gr} _F^p(W^q/W^{q+2}(K)) \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \rar \dar
& F^p/F^{p+2} \operatorname {Gr} _W^q K \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^q K \rar \dar
& 0
\\ & 0
& 0
& 0
\end {tikzcd}
The exterior rows and columns of this square define connection morphisms
\begin {aligned} {}_{F,W}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \\ {}_{W,F}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1} K. \end {aligned}
These morphisms satisfy
\begin {gathered} {}_{FW}d_1 \circ {}_{WF}d_1 + {}_{WF}d_1 \circ {}_{FW}d_1 = 0 \\ {}_{FW}d_1^2 = 0 \\ {}_{WF}d_1^2 = 0 \end {gathered}
The morphisms {}_{FW}d_1 are the morphisms d_1 of the spectral sequences E^{(q)}, with the E_1^{(q)p,n-p} term equal to
1008Equationhodge-theory-ii-1.4.9.1hodge-theory-ii-1.4.9.1.xml1.4.9.1hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _W^q K) = {}_WE_1^{q,n-q} \tag{1.4.9.1}
defined by the filtered complex \operatorname {Gr} _W^q(K).
This spectral sequence abuts to the filtration induced by F on \operatorname {H} ^ \bullet \operatorname {Gr} _W^q K.
Similarly, the {}_{WF}d_1 are the d_1 of spectral sequences with the same initial terms
1009Equationhodge-theory-ii-1.4.9.2hodge-theory-ii-1.4.9.2.xml1.4.9.2hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _F^p K). \tag{1.4.9.1}
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& \operatorname {H} ^n \operatorname {Gr} _W^q K
\ar [Rightarrow,dr,"{}_Wd_1"]
\\ \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K
\ar [Rightarrow,ur,"{}_{F,W}d_1"]
\ar [Rightarrow,dr,swap,"{}_{W,F}d_1"]
&& \operatorname {H} ^n K
\\ & \operatorname {H} ^n \operatorname {Gr} _F^p K
\ar [Rightarrow,ur,swap,"{}_Fd_1"]
\end {tikzcd}
These constructions are symmetric in F and W via the isomorphism
\operatorname {Gr} _F^p \operatorname {Gr} _W^q \sim \operatorname {Gr} _W^q \operatorname {Gr} _F^p. 1011hodge-theory-ii-1.4.10hodge-theory-ii-1.4.10.xml1.4.10hodge-theory-ii-1.4
We can also interpret the {}_{W,F}d_1 are the initial morphisms of a morphism of spectral sequences , abutting to {}_Wd_1.
Indeed, let C^q be the cone of the morphism
W^q(K)/W^{q+2}(K) \to \operatorname {Gr} _W^q(K).
In the diagram
\Sigma \colon \operatorname {Gr} _W^{q+1}(K)[1] \xrightarrow {u} C^q \xleftarrow {i} \operatorname {Gr} _W^q(K)
the morphism u is a quasi-isomorphism, and we have
{}_Wd_1 = \operatorname {H} (u)^{-1} \circ \operatorname {H} (i).
In fact, u is even a filtered (for F) quasi-isomorphism, and the above construction defines a morphism from the spectral sequence defined by ( \operatorname {Gr} _W^q(K),F) to that defined by ( \operatorname {Gr} _W^{q+1}(K)[1],F), and it abuts to {}_Wd_1.
The initial term of this morphism, induced by \operatorname {Gr} _F( \Sigma ) is exactly {}_{W,F}d_1.
1012hodge-theory-ii-1.4.11hodge-theory-ii-1.4.11.xml1.4.11hodge-theory-ii-1.4
These constructions pass as they are to hypercohomology.
Let K be a complex endowed with two biregular filtrations F and W.
A bifiltered T-acyclic resolution of K is a bifiltered quasi-isomorphism i \colon K \to K' such that the \operatorname {Gr} _F^p \operatorname {Gr} _W^n({K'}^m) are T-acyclic.
Such a morphism always exists.
In the particular case where \mathscr {A} is the category of sheaves of A-modules on a topological space X, and where T is the functor \Gamma from \mathscr {A} to the category of A-modules, then an example of a bifiltered T-acyclic resolution of K is the simple complex associated to the double complex given by the Godement resolution \mathscr {C}^ \bullet (K) of K, filtered by the \mathscr {C}^ \bullet (F^p(K)) and the \mathscr {C}^ \bullet (W^n(K)).
Since \mathscr {C}^ \bullet is exact, we have
\operatorname {Gr} _F \operatorname {Gr} _W( \mathscr {C}^ \bullet (K)) \simeq \mathscr {C}^ \bullet ( \operatorname {Gr} _F \operatorname {Gr} _W(K)).
We will not have need here of any other case.
If K' is a bifiltered T-acyclic resolution of K, then the complex TK' is filtered by the TF^pK' and the TW^qK' ().
Furthermore, \operatorname {Gr} _W^n(K') is a T-acyclic filtered (for F) resolution of \operatorname {Gr} _F^n(K), and \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K') is a T-acyclic resolution of \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K), and
\begin {gathered} T \operatorname {Gr} _F K' \approx \operatorname {Gr} _F TK' \qquad \text {as }W \text {-filtered complexes} \\ T \operatorname {Gr} _W K' \approx \operatorname {Gr} _W TK' \qquad \text {as }F \text {-filtered complexes} \\ T \operatorname {Gr} _F \operatorname {Gr} _W K' \approx \operatorname {Gr} _F \operatorname {Gr} _W TK'. \end {gathered} 1013Lemmahodge-theory-ii-1.4.12hodge-theory-ii-1.4.12.xml1.4.12hodge-theory-ii-1.4
Under the hypotheses of :
The initial terms of the hypercohomology spectral sequences
{}_WE_1^{q,n-q} = \mathrm {R} ^nT( \operatorname {Gr} _W^qK) \Rightarrow \mathrm {R} ^nT(K) \tag{1}
{}_FE_1^{p,n-p} = \mathrm {R} ^nT( \operatorname {Gr} _F^pK) \Rightarrow \mathrm {R} ^nT(K) \tag{2}
are abutments of the hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _W^qK and \operatorname {Gr} _F^pK, with E_1 pages given by
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_WE_1^{q,n-q} \qquad \text {for fixed }q \tag{3}
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_FE_1^{p,n-p} \qquad \text {for fixed }p \tag{4}
The filtration of {}_WE_1^{p,n-p}, abutting to the spectral sequence (3), is the filtration of {}_WE_1^{p,n-p}(TK') induced by the filtration F of TK'.
For the differentials of the complexes \operatorname {Gr} _W^n(T(K')) to be strictly compatible with the filtration F, it is necessary and sufficient that the hypercohomology spectral sequences (3) degenerate on the E_1 page.
The morphisms d_1 of the spectral sequence (4) are the initial terms of the degree-1 morphisms of the spectral sequences (3) that abut to the morphisms d_1 of the spectral sequence (1).
895Proofhodge-theory-ii-1.4.12
Claims (i) and (iv) follow from and applied to TK' via the isomorphisms .
Claim (ii) is then trivial by the definition of the recurrent filtration F (identical to the discrete filtrations by and (iii) of ), and claim (iii) follows from .
1642hodge-theory-ii-2hodge-theory-ii-2.xmlHodge structures2hodge-theory-ii1639hodge-theory-ii-2.1hodge-theory-ii-2.1.xmlPure structures2.1hodge-theory-ii-2773hodge-theory-ii-2.1.1hodge-theory-ii-2.1.1.xml2.1.1hodge-theory-ii-2.1
In all the following, we denote by \mathbb {C} an algebraic closure of \mathbb {R}, and we do not suppose to have chosen a root i of the equation x^2+1=0.
The theory will be invariant under complex conjugation (cf. [hodge-theory-ii-2.1.14 (?)]).
774hodge-theory-ii-2.1.2hodge-theory-ii-2.1.2.xml2.1.2hodge-theory-ii-2.1
We denote by S the real algebraic group \mathbb {C} ^*, given by restriction of scalars à la Weil from \mathbb {C} to \mathbb {R} of the group \mathbb {G}_ \mathrm {m}:
\begin {aligned} S &= \prod _{ \mathbb {C} {/} \mathbb {R} } \mathbb {G}_ \mathrm {m} \\ S( \mathbb {R} ) &= \mathbb {C} ^*. \end {aligned}
The group S is a torus, i.e. it is connected and of multiplicative type.
It is thus described by the free abelian group of finite type
X(S) = \operatorname {Hom} (S_ \mathbb {C} , \mathbb {G}_ \mathrm {m} ) = \underline { \operatorname {Hom} } (S, \mathbb {G}_ \mathrm {m} )( \mathbb {C} )
of its complex characters, endowed with the action of \operatorname {Gal} ( \mathbb {C} {/} \mathbb {R} )= \mathbb {Z} {/}(2).
The group X(S) has generators z and \bar {z}, which induce (respectively) the identity and complex conjugation:
\mathbb {C} ^* = S( \mathbb {R} ) \to S( \mathbb {C} ) \to \mathbb {G}_ \mathrm {m} ( \mathbb {C} ) = \mathbb {C} ^*.
Complex conjugation exchanges z and \bar {z}.
780hodge-theory-ii-2.1.3hodge-theory-ii-2.1.3.xml2.1.3hodge-theory-ii-2.1
We have a canonical map
775Equationhodge-theory-ii-2.1.3.1hodge-theory-ii-2.1.3.1.xml2.1.3.1hodge-theory-ii-2.1.3 w \colon \mathbb {G}_ \mathrm {m} \to S \tag{2.1.3.1}
that, on the real points, induces the inclusion of \mathbb {R} ^* into \mathbb {C} ^*.
We have
776Equationhodge-theory-ii-2.1.3.2hodge-theory-ii-2.1.3.2.xml2.1.3.2hodge-theory-ii-2.1.3 zw = \bar {z}w = \mathrm {Id} . \tag{2.1.3.2}
We also have a map
777Equationhodge-theory-ii-2.1.3.3hodge-theory-ii-2.1.3.3.xml2.1.3.3hodge-theory-ii-2.1.3 N \colon S \to \mathbb {G}_ \mathrm {m} \tag{2.1.3.3}
that on the real points can be identified with the norm N_{ \mathbb {C} {/} \mathbb {R} } \colon \mathbb {C} ^* \to \mathbb {R} ^*.
We have
778Equationhodge-theory-ii-2.1.3.4hodge-theory-ii-2.1.3.4.xml2.1.3.4hodge-theory-ii-2.1.3 N = z \bar {z} \tag{2.1.3.4}
779Equationhodge-theory-ii-2.1.3.5hodge-theory-ii-2.1.3.5.xml2.1.3.5hodge-theory-ii-2.1.3 N \circ w = (x \mapsto x^2). \tag{2.1.3.5} 781Definitionhodge-theory-ii-2.1.4hodge-theory-ii-2.1.4.xml2.1.4hodge-theory-ii-2.1
A real Hodge structure is a real vector space V of finite dimension endowed with an action of the real algebraic group S.
783hodge-theory-ii-2.1.5hodge-theory-ii-2.1.5.xml2.1.5hodge-theory-ii-2.1
By the general theory of groups of multiplicative type, giving a real Hodge structure on V is equivalent to giving a bigrading V^{p,q} of V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that satisfies \overline {V^{pq}}=V^{qp}.
The action of S and the bigrading are mutually determined via the condition:
782hodge-theory-ii-2.1.5.1hodge-theory-ii-2.1.5.1.xml2.1.5.1hodge-theory-ii-2.1.5
On V^{pq}, S acts by multiplication by z^p \bar {z}^q.
784hodge-theory-ii-2.1.6hodge-theory-ii-2.1.6.xml2.1.6hodge-theory-ii-2.1
Let ( \mathbb {C} , \times ) be the multiplicative monoid, and let
\bar {S} = \prod _{ \mathbb {C} {/} \mathbb {R} } ( \mathbb {C} , \times ).
If V is a real vector space, then we can show that it is equivalent to either give an action of \bar {S} on V or to give a bigrading V^{pq} on V such that \overline {V^{pq}}=V^{qp} and V^{pq}=0 for p<0 or q<0.
785hodge-theory-ii-2.1.7hodge-theory-ii-2.1.7.xml2.1.7hodge-theory-ii-2.1
Let V be a real Hodge structure, defined by a representation \sigma of S and a bigrading V^{pq}.
The grading of V_ \mathbb {C} by the V_ \mathbb {C} ^n= \sum _{p+q=n}V^{pq} is then defined over \mathbb {R}.
We call this the weight grading.
On V^n=V \cap V_ \mathbb {C} ^n, the representation \sigma w of \mathbb {G}_ \mathrm {m} is multiplication by x^n.
We say that V is of weight n if V^{pq}=0 for p+q \neq n, i.e. if \sigma w is multiplication by x^n.
786hodge-theory-ii-2.1.8hodge-theory-ii-2.1.8.xml2.1.8hodge-theory-ii-2.1
Let V be a real Hodge structure.
The Hodge filtration on V_ \mathbb {C} is defined by
F^p(V_ \mathbb {C} ) = \sum _{p' \geqslant p} V^{p'q'}.
By , we have:
790Propositionhodge-theory-ii-2.1.9hodge-theory-ii-2.1.9.xml2.1.9hodge-theory-ii-2.1
Let n be an integer.
The construction establishes an equivalence of categories between:
the category of real Hodge structures of weight n;
the category of pairs consisting of a real vector space V of finite dimension and of a filtration F on V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that is n-opposite to its complex conjugate \bar {F}.
794Definitionhodge-theory-ii-2.1.10hodge-theory-ii-2.1.10.xml2.1.10hodge-theory-ii-2.1
A Hodge structure H, of weight n, consists of
a \mathbb {Z}-module H_ \mathbb {Z} of finite type (the "integral lattice");
a real Hodge structure of weight n on H_ \mathbb {R} = \mathbb {R} \otimes _ \mathbb {Z} H_ \mathbb {Z}.
795hodge-theory-ii-2.1.11hodge-theory-ii-2.1.11.xml2.1.11hodge-theory-ii-2.1
A morphism f \colon H \to H' is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} such that f_ \mathbb {R} \colon H_ \mathbb {R} \to H'_ \mathbb {R} is compatible with the action of S (i.e. such that f_ \mathbb {C} is compatible with the bigrading, or with the Hodge filtration).
Hodge structures of weight n form an abelian category.
If H is of weight n, and H' is of weight n', then we define a Hodge structure H \otimes H' of weight n+n' by the equations:
(H \otimes H')_ \mathbb {Z} =H_ \mathbb {Z} \otimes H'_ \mathbb {Z};
the action of S on (H \otimes H')_ \mathbb {R} =H_ \mathbb {R} \otimes H'_ \mathbb {R} is the tensor product of the actions of S on H_ \mathbb {R} and on H'_ \mathbb {R}.
The bigrading (resp. Hodge filtration) of (H \otimes H')_ \mathbb {C} =H_ \mathbb {C} \otimes H'_ \mathbb {C} is the tensor product of the bigradings (resp. Hodge filtrations (cf. )) of H_ \mathbb {C} and of H'_ \mathbb {C}.
We define in an analogous manner the Hodge structure \underline { \operatorname {Hom} } (H,H') (of weight n'-n), the Hodge structures \wedge ^p H (of weight pn), and the dual Hodge structure H^*.
The internal hom \underline { \operatorname {Hom} } above and the homomorphism group, are related by:
620Remarkhodge-theory-ii-2.1.11.1hodge-theory-ii-2.1.11.1.xml2.1.11.1hodge-theory-ii-2.1.11\operatorname {Hom} (H,H') is the subgroup of
\underline { \operatorname {Hom} } (H,H')_ \mathbb {Z} = \operatorname {Hom} _ \mathbb {Z} (H_ \mathbb {Z} ,H'_ \mathbb {Z} )
consisting of elements of degree (0,0).
The actions of S on H_ \mathbb {R}, H'_ \mathbb {R}, and \underline { \operatorname {Hom} } (H,H')_ \mathbb {R} = \operatorname {Hom} (H_ \mathbb {R} ,H'_ \mathbb {R} ) are related by
s(f(x)) = s(f)(s(x)).
This means that if f is of degree (0,0), i.e. invariant under S, then it commutes with the action of S.
796hodge-theory-ii-2.1.12hodge-theory-ii-2.1.12.xml2.1.12hodge-theory-ii-2.1
If A is a Noetherian subring of \mathbb {R}, then we define a Hodge A-structure of weight n to consist of an A-module H_A of finite type along with a real Hodge structure of weight n on H_ \mathbb {R} =H_A \otimes _A \mathbb {R}.
This definition is mostly used for A= \mathbb {Q}.
A Hodge A-structure consists of an A-module H_A of finite type along with a real Hodge structure on H_ \mathbb {R} =H_A \otimes _A \mathbb {R} such that the weight grading is defined over the field of fractions of A.
797Definitionhodge-theory-ii-2.1.13hodge-theory-ii-2.1.13.xml2.1.13hodge-theory-ii-2.1
The Tate Hodge structure \mathbb {Z} (1) is the Hodge structure of weight -2, of rank 1, of pure bidegree (-1,-1), with integral lattice 2 \pi i \mathbb {Z} \subset \mathbb {C}.
The action of S is thus given by multiplication with the inverse of the norm ().
For n \in \mathbb {Z}, we define \mathbb {Z} (n) as the n-th tensor power of \mathbb {Z} (1), so \mathbb {Z} (n) is the Hodge structure of weight -2n, of rank 1, of pure bidegree (-n,-n), with integral lattice (2 \pi i)^n \mathbb {Z} \subset \mathbb {C}.
THe action of S is multiplication by N(x)^{-n}.
798hodge-theory-ii-2.1.14hodge-theory-ii-2.1.14.xml2.1.14hodge-theory-ii-2.1
The choice in \mathbb {C} of a solution i of the equation x^2+1=0 determines, on each complex variety X of pure dimension n, an orientation \operatorname {or} _i(X).
Replace i by -i gives
\operatorname {or} _{-i}(X) = (-1)^n \operatorname {or} _i(X).
The choice of i also defines an element C of order 4 in S( \mathbb {R} ), given by the image of i under the isomorphism S( \mathbb {R} ) \simeq \mathbb {C} ^*.
Finally, it also defines an isomorphism between \mathbb {Z} and the integral lattice of \mathbb {Z} (n), given by multiplication by (2 \pi i)^n.
When i, an orientation of X, C, or an identification \mathbb {Z} \sim \mathbb {Z} (n)_ \mathbb {Z} appear in an equation, it is implicitly understood that they all follow from a single choice of the same i, and that by replacing i with -i we obtain an equivalent equation.
799Definitionhodge-theory-ii-2.1.15hodge-theory-ii-2.1.15.xml2.1.15hodge-theory-ii-2.1
A polarisation of a Hodge structure H of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {Z} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
800hodge-theory-ii-2.1.16hodge-theory-ii-2.1.16.xml2.1.16hodge-theory-ii-2.1
The real Tate Hodge structure is the real Hodge structure \mathbb {R} (1) underlying \mathbb {Z} (1).
We similarly define \mathbb {R} (n) as underlying \mathbb {Z} (1).
A polarisation of a real Hodge structure of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {R} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
A polarisation is entirely defined by the positive definite quadratic form (2 \pi i)^n(x,Cy) on H_ \mathbb {R}, imposed with only the condition of being invariant under the compact sub-torus of S given by the kernel of N.
We have
(x,y) = (Cx,Cy) = (y,C^2x) = (-1)^n(y,x).
The form (x,y) is thus symmetric or alternating, depending on the parity of n.
801hodge-theory-ii-2.1.17hodge-theory-ii-2.1.17.xml2.1.17hodge-theory-ii-2.1
The reader can generalise these definitions to Hodge A-structures of weight n ().
1640hodge-theory-ii-2.2hodge-theory-ii-2.2.xmlHodge theory2.2hodge-theory-ii-21174hodge-theory-ii-2.2.1hodge-theory-ii-2.2.1.xml2.2.1hodge-theory-ii-2.2
Let X be a compact Kähler variety (for example, smooth and projective).
By the holomorphic Poincaré lemma, the de Rham complex \Omega _X^ \bullet is a resolution of the constant sheaf \mathbb {C}.
We thus have an isomorphism ()
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \sim \operatorname { \mathbb {H}} ^ \bullet (X, \Omega _X^ \bullet )
and the stupid filtration on \Omega _X^ \bullet () defines the hypercohomology spectral sequence
388Equationhodge-theory-ii-2.2.1.1hodge-theory-ii-2.2.1.1.xml2.2.1.1hodge-theory-ii-2.2.1 E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^q) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} ) \tag{2.2.1.1}
which abuts to the Hodge filtration of \operatorname {H} ^ \bullet (X, \mathbb {C} ).
By Hodge theory [H1952; W1958], we have:
The spectral sequence degenerates: E_1=E_ \infty.
The Hodge filtration of \operatorname {H} ^n(X, \mathbb {C} ) is n-opposite to the complex conjugate filtration.
1175hodge-theory-ii-2.2.2hodge-theory-ii-2.2.2.xml2.2.2hodge-theory-ii-2.2
Let V be a local system of real vector spaces over X, i.e. a sheaf of \mathbb {R}-vector spaces that is locally isomorphic to a constant sheaf \mathbb {R} ^n.
Suppose that there exists on V a bilinear form
Q \colon V \otimes V \to \mathbb {R}
that is locally constant and defined.
If X is connected, this is the case if V is defined by a representation of a finite quotient of the fundamental group of X.
Statements (A) and (B) above remain true as they are, without needing to change the proofs, for cohomology with coefficients in V_ \mathbb {C} =V \otimes _ \mathbb {R} \mathbb {C}, since the spectral sequence
E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^p(V)) \Rightarrow \operatorname {H} ^{p+q}(X,V_ \mathbb {C} )
induced by the de Rham resolution of V_ \mathbb {C} by \Omega _X^ \bullet (V_ \mathbb {C} ) degenerates, and abuts to a filtration on \operatorname {H} ^n(X,V_ \mathbb {C} ) that is n-opposite to the complex conjugate filtration.
The vector space \operatorname {H} ^n(X,V) is thus endowed with a canonical real Hodge structure of weight n.
1176hodge-theory-ii-2.2.3hodge-theory-ii-2.2.3.xml2.2.3hodge-theory-ii-2.2
We show in [D1968] that the statements in remain true when X is a non-singular complete algebraic, not necessarily Kähler, variety.
The proof from loc. cit., based on a reduction to the projective case by Chow's lemma and resolution of singularities, extends to the case of .
1179hodge-theory-ii-2.2.4hodge-theory-ii-2.2.4.xml2.2.4hodge-theory-ii-2.2
Let \mathcal {L} be an invertible sheaf.
Here are two ways of defining the class c_1( \mathcal {L}).
1177hodge-theory-ii-2.2.4.1hodge-theory-ii-2.2.4.1.xml2.2.4.1hodge-theory-ii-2.2.4
The sheaf \mathcal {L} defines an element c in \operatorname {H} ^1(X, \mathcal {O} ^*).
Its image under \mathrm {d} f/f \colon \mathcal {O} ^* \to \Omega ^1 lies in \operatorname {H} ^1(X, \Omega ^1).
More precisely, \mathrm {d} f/f defines a morphism of complexes
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to (0 \to \Omega _X^1 \to \Omega _X^2 \to \ldots ) = \sigma _{ \geqslant1 }( \Omega _X^ \bullet ).
This complex maps to \Omega _X^ \bullet, whence
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to \Omega _X^ \bullet
and the image of c under \mathrm {d} \log is in \operatorname { \mathbb {H}} ^2( \Omega _X^ \bullet ).
This construction still makes sense for an algebraic variety over an arbitrary field k.
For k= \mathbb {C}, we further have
\operatorname {H} ^2(X, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^2(X, \Omega _X^ \bullet )
whence a class
c'_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {C} ). 1178hodge-theory-ii-2.2.4.2hodge-theory-ii-2.2.4.2.xml2.2.4.2hodge-theory-ii-2.2.4
The exponential exact sequence
0 \to \mathbb {Z} (1) \to \mathcal {O} \to \mathcal {O} ^* \to 0
defines a homomorphism
\partial \colon \operatorname {H} ^1(X, \mathcal {O} ^*) \to \operatorname {H} ^2(X, \mathbb {Z} (1))
whence a class
\partial c = c''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} (1)).
If we have a choice of i, then this class is identified with
c'''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} )
and for \mathcal {L}= \mathcal {O} (D), this c'''_1( \mathcal {L}) is exactly the integer cohomology class defined by D (with the orientations being defined by i).
1182hodge-theory-ii-2.2.5hodge-theory-ii-2.2.5.xml2.2.5hodge-theory-ii-2.2
We will prove:
1180hodge-theory-ii-2.2.5.1hodge-theory-ii-2.2.5.1.xml2.2.5.1hodge-theory-ii-2.2.5
For \alpha the natural injection of \mathbb {Z} (1) into \mathbb {C}, we have
- \alpha c''_1( \mathcal {L}) = c'_1( \mathcal {L}).
1181hodge-theory-ii-2.2.5.2hodge-theory-ii-2.2.5.2.xml2.2.5.2hodge-theory-ii-2.2.5
For \bta the natural injection of \mathbb {Z} into \mathbb {C}, we have
- \beta c'''_1( \mathcal {L}) = \frac {1}{2 \pi i}c'_1( \mathcal {L}).
We prove this by considering the following diagram of complexes, in which \approx denotes a quasi-isomorphism, and whose upper triangle anti-commutes up to homotopy:
\begin {CD} \mathbb {C} @> \approx >> \Omega _X^ \bullet @<<< \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @| @. @| \\ \mathbb {C} @<<< ( \mathbb {C} \to \mathcal {O} ) @> \approx > \mathrm {d} > \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @AAA @AAA @AA{ \mathrm {d} \log }A \\ \mathbb {Z} (1) @<<< ( \mathbb {Z} (1) \to \mathcal {O} ) @> \approx > \exp > \mathcal {O} ^*[-1] \end {CD} 1183hodge-theory-ii-2.2.6hodge-theory-ii-2.2.6.xml2.2.6hodge-theory-ii-2.2
Let X be a non-singular projective variety of pure dimension n.
A choice of i defines an orientation of X and an isomorphism \mathbb {Z} (1) \simeq \mathbb {Z}.
The corresponding trace morphism
\operatorname {H} ^{2n}(X, \mathbb {Z} (n)) \to \mathbb {Z}
that is induced does not depend on the choice of i.
By Hodge, for i \leqslant n, the morphism
L^{n-i} = - \wedge c''_1( \mathcal {O} (1))^{n-i} \colon \operatorname {H} ^i(X, \mathbb {Z} ) \to \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i))
is an isomorphism, and, combined with Poincaré duality
\operatorname {H} ^i(X, \mathbb {Z} ) \otimes \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i)) \to \operatorname {H} ^{2n}(X, \mathbb {Z} (n-i)) \to \mathbb {Z} (-i)
gives a polarisation on the primitive part \operatorname {Ker} (L^{n-i+1}) of \operatorname {H} ^i(X, \mathbb {Z} ).
We thus deduce that the rational Hodge structures \operatorname {H} ^i(X, \mathbb {Q} ) are polarisable.
1641hodge-theory-ii-2.3hodge-theory-ii-2.3.xmlMixed structures2.3hodge-theory-ii-21218Definitionhodge-theory-ii-2.3.1hodge-theory-ii-2.3.1.xml2.3.1hodge-theory-ii-2.3
A mixed Hodge structure H consists of
A \mathbb {Z}-module H_ \mathbb {Z} of finite type, called the "integral lattice";
A finite increasing filtration W_n on H_ \mathbb {Q} = \mathbb {Q} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the weight filtration;
A finite filtration F^p on H_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the Hodge filtration.
We demand that, on H_ \mathbb {C}, the filtration W_ \mathbb {C} induced by extension of scalars of W, the filtration F, and the complex conjugate \bar {F} form a system (W_ \mathbb {C} ,F, \bar {F}) of three opposite filtrations ( and ).
1219hodge-theory-ii-2.3.2hodge-theory-ii-2.3.2.xml2.3.2hodge-theory-ii-2.3
We also denote by W the filtration of H_ \mathbb {Z} given by the inverse image of the filtration W on H_ \mathbb {Q}.
The axiom of mixed Hodge structures implies that, for each n, the filtration F induces on \mathbb {C} \otimes _ \mathbb {Z} \operatorname {Gr} _n^W(H_ \mathbb {Z} ) a filtration that is n-opposite to its complex conjugate.
By , \operatorname {Gr} _n^W(H_ \mathbb {Z} ) is endowed with a Hodge structure of weight n, with Hodge filtration induced by F.
1220Examplehodge-theory-ii-2.3.3hodge-theory-ii-2.3.3.xml2.3.3hodge-theory-ii-2.3
If H is a Hodge structure of weight n, then we define a mixed Hodge structure with the same integral lattice and the same Hodge filtration by setting
W_i(H_ \mathbb {Q} ) = \begin {cases} 0 & \text {for } i<n \\ H_ \mathbb {Q} & \text {for } i \geqslant n. \end {cases} 1221hodge-theory-ii-2.3.4hodge-theory-ii-2.3.4.xml2.3.4hodge-theory-ii-2.3
A morphism f \colon H \to H' of Hodge structures is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} that is compatible with the filtrations W and F (and thus compatible with \bar {F}).
We immediately deduce from the following theorem.
1222Theoremhodge-theory-ii-2.3.5hodge-theory-ii-2.3.5.xml2.3.5hodge-theory-ii-2.3
The category of mixed Hodge structures is abelian.
The integral lattice of the kernel (resp. cokernel) of a morphism f \colon H \to H' is the kernel (resp. cokernel) K of f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z}, with K \otimes \mathbb {Q} and K \otimes \mathbb {C} being endowed with the induced filtrations (resp. quotient filtrations) of the filtrations W and F of H_ \mathbb {Q} and H_ \mathbb {C} (resp. of H'_ \mathbb {Q} and H'_ \mathbb {C}).
Every morphism f \colon H \to H' is strictly compatible with the filtration W of H_ \mathbb {Q} and H'_ \mathbb {Q} and with the filtration F of H_ \mathbb {C} and H'_ \mathbb {C}.
It induces morphisms of Hodge \mathbb {Q}-structures
\operatorname {Gr} _n^w(f) \colon \operatorname {Gr} _n^W(H_ \mathbb {Q} ) \to \operatorname {Gr} _n^W(H'_ \mathbb {Q} ).
It also induces morphisms
\operatorname {Gr} _F^p(F) \colon \operatorname {Gr} _F^p(H_ \mathbb {C} ) \to \operatorname {Gr} _F^p(H'_ \mathbb {C} )
that are strictly compatible with the filtration induced by W_ \mathbb {C}.
The functor \operatorname {Gr} _n^W is an exact functor from the category of mixed Hodge structures to the category of Hodge \mathbb {Q}-structures of weight n.
The functor \operatorname {Gr} _F^p is an exact functor.
1223hodge-theory-ii-2.3.6hodge-theory-ii-2.3.6.xml2.3.6hodge-theory-ii-2.3
Let H be a mixed Hodge structure.
The W_n(H_ \mathbb {Z} ), endowed with the filtrations induced by W and F, then form mixed Hodge substructures W_n(H) of H.
The quotient W_n(H)/W_{n-1}(H) can be identified with \operatorname {Gr} _n^W(H_ \mathbb {Z} ) endowed with its mixed Hodge structure ( and ).
We denote this Hodge structure by \operatorname {Gr} _n^W(H).
1224hodge-theory-ii-2.3.7hodge-theory-ii-2.3.7.xml2.3.7hodge-theory-ii-2.3
We set
H^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _{p+q}^W(H_ \mathbb {C} ) = ( \operatorname {Gr} _{p+q}^W(H))^{p,q}.
The Hodge numbers of H are the integers
h^{p,q} = \dim _ \mathbb {C} H^{p,q}.
The Hodge number h^{pq} of H is thus the Hodge number h^{pq} of the Hodge structure \operatorname {Gr} _{p+q}^W(H)1225hodge-theory-ii-2.3.8hodge-theory-ii-2.3.8.xml2.3.8hodge-theory-ii-2.3
We define a mixed Hodge \mathbb {Q}-structure H to consist of a vector space H_ \mathbb {Q} of finite dimension over \mathbb {Q}, a finite increasing filtration W on H_ \mathbb {Q}, and a finite decreasing filtration F on H_ \mathbb {C}, with the filtrations W_ \mathbb {C}, F, and \bar {F} being opposite.
generalises trivially to this variation.
1645hodge-theory-ii-3hodge-theory-ii-3.xmlHodge theory of non-singular algebraic varieties3hodge-theory-ii1643hodge-theory-ii-3.1hodge-theory-ii-3.1.xmlLogarithmic poles and residues3.1hodge-theory-ii-31254hodge-theory-ii-3.1.1hodge-theory-ii-3.1.1.xml3.1.1hodge-theory-ii-3.1
We recall several classical properties of "logarithmic poles" of holomorphic differential forms.
The reader will find proofs in [D1970, II, (3.1) to (3.7)], for example.
1255hodge-theory-ii-3.1.2hodge-theory-ii-3.1.2.xml3.1.2hodge-theory-ii-3.1
A divisor Y in a smooth complex analytic variety X is said to be of normal crossing if the inclusion of Y into X is locally isomorphic to the inclusion of a union of coordinate hyperplanes in \mathbb {C} ^n;
this does not imply that Y is the union of smooth divisors.
Let Y be a normal crossing divisor in X, and j the inclusion of X^*=X \setminus Y into X.
We denote by \Omega _X^1 \langle Y \rangle the locally free sub-\mathcal {O}-module of j_* \Omega _{X^*}^1 generated by \Omega _X^1 and the \mathrm {d} z_i/z_i for z_i local coordinates of a local irreducible component of Y.
The sheaf \Omega _X^p \langle Y \rangle of differential p-forms on X with logarithmic poles along Y is by the definition the locally free sub-sheaf \wedge ^p \Omega _X^1 \langle Y \rangle of j_* \Omega _{X^*}^p.
1259Propositionhodge-theory-ii-3.1.3hodge-theory-ii-3.1.3.xml3.1.3hodge-theory-ii-3.1
A section \alpha of j_* \Omega _{X^*}^p belongs to \Omega _X^p \langle Y \rangle if and only if \alpha and \mathrm {d} \alpha have at worst simple poles along the divisor Y.
The \Omega _X^p \langle Y \rangle form the smallest subcomplex of j_* \Omega _{X^*}^ \bullet that is stable under exterior product and that contains \Omega _X^ \bullet and the logarithmic differential \mathrm {d} f/f of every local section of j_* \Omega _{X^*}^p meromorphic along Y.
We call \Omega _X^ \bullet \langle Y \rangle the logarithmic de Rham complex of X along Y.
By (ii) of , this complex is contravariant in the pair (X,X^*).
1261hodge-theory-ii-3.1.4hodge-theory-ii-3.1.4.xml3.1.4hodge-theory-ii-3.1
Locally on X, Y is the union of various smooth divisors Y_i, and we denote by Y^n (resp. \widetilde {Y}^n) the union (resp. disjoint sum) of the n-fold intersections of the Y_i.
The Y^n glue to give a subspace Y^n of X, and the \widetilde {Y}^n glue to give the normalisation variety of Y^n.
We have \widetilde {Y}^0=Y^0=X, and we set \widetilde {Y}= \widetilde {Y}^1.
We define the two-element set of orientations of an n-element set E as the set of generators of \wedge ^n \mathbb {Z} ^E.
For n \geqslant2, this set is that of the conjugation classes under the alternating group of total orders on E.
If, to each point y of \widetilde {Y}^n, we associate the set of n local components of Y that contain the image in X of a neighbourhood of y in \widetilde {Y}^n, then we define on \widetilde {Y}^n a local system E_n of n-element sets.
The local system of orientations of these sets is a \mathbb {Z} {/}(2)-torsor.
This torsor defines, via the inclusion of \mathbb {Z} {/}(2) into \mathbb {C} ^*, a complex local system \varepsilon ^n of rank 1 on \widetilde {Y}^n, endowed with an isomorphism ( \varepsilon ^n)^{ \otimes2 } \simeq \mathbb {C}.
We have
\varepsilon ^n \simeq \bigwedge ^n \mathbb {C} ^{E_n}.
Locally on \widetilde {Y}^n, \varepsilon ^n is endowed with two opposite isomorphisms \pm \alpha \colon \varepsilon ^n \xrightarrow { \sim } \mathbb {C}.
We set
\varepsilon _ \mathbb {Z} ^n = \alpha ^{-1}((2 \pi i)^{-n} \mathbb {Z} ).
There is a way of seeing \varepsilon ^n, endowed with \varepsilon _ \mathbb {Z} ^n, as a twisted version of \mathbb {Z} (-n) over \widetilde {Y}^n.
We denote by \varepsilon _X^n (resp. by ( \varepsilon _X^n)_ \mathbb {Z}) the direct image of \varepsilon ^n (resp. of \varepsilon _ \mathbb {Z} ^n) under the mat from \widetilde {Y}^n to X.
We have
1260Equationhodge-theory-ii-3.1.4.1hodge-theory-ii-3.1.4.1.xml3.1.4.1hodge-theory-ii-3.1.4 \varepsilon _X^n \simeq \bigwedge ^n \varepsilon _X^1 \qquad \text {for }n \geqslant0 . \tag{3.1.4.1}
If Y is the union of distinct smooth divisors (Y_i)_{i \in I}, then the choice of a total order on I trivialises the \varepsilon ^n.
1263hodge-theory-ii-3.1.5hodge-theory-ii-3.1.5.xml3.1.5hodge-theory-ii-3.1
We denote by W_n( \Omega _X^p \langle Y \rangle ) the sub-module of \Omega _X^p \langle Y \rangle consisting of linear combinations of products
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \qquad \text {for }m \leqslant n
with \alpha holomorphic and the t_{i(j)} local coordinates of the distinct local components Y_j of Y.
We define the weight filtration of \Omega _X^ \bullet \langle Y \rangle to be the increasing filtration by the subcomplexes W_n( \Omega _X^ \bullet \langle Y \rangle ).
We have
1262Equationhodge-theory-ii-3.1.5.1hodge-theory-ii-3.1.5.1.xml3.1.5.1hodge-theory-ii-3.1.5 W_n( \Omega _X^p \langle Y \rangle ) \wedge W_m( \Omega _X^p \langle Y \rangle ) \subset W_{n+m}( \Omega _X^{p+q} \langle Y \rangle ). \tag{3.1.5.1}
If we denote by i_n the map from \widetilde {Y}^n to X, then we can show that the mapping
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \longmapsto ( \alpha |Y_{i(1)} \cap \ldots \cap Y_{i(n)}) \otimes ( \text {orientation }i(1) \ldots i(n))
defines an isomorphism of complexes
\operatorname {Res} \colon \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle ) \approx (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n]
(the Poincaré residue).
1264hodge-theory-ii-3.1.6hodge-theory-ii-3.1.6.xml3.1.6hodge-theory-ii-3.1
The interpretation that follows, in terms of [hodge-theory-ii-3.1.5.2 (?)], of the Leray spectral sequence for the inclusion of X^* into X, was pointed out to me by N. Katz.
It will allow us to prove a point that I had initially considered as evident (the first part of (ii) of [hodge-theory-ii-3.2.5 (?)]).
1265hodge-theory-ii-3.1.7hodge-theory-ii-3.1.7.xml3.1.7hodge-theory-ii-3.1
Every point of X admits a fundamental system of Stein open neighbourhoods whose intersections on X^* are again Stein.
For a coherent analytic sheaf \mathscr { F } on X^*, we thus have \mathrm {R} ^i j_* \mathscr { F } =0 for i>0.
The de Rham complex \Omega _{X^*}^ \bullet is thus a resolution of the constant sheaf \mathbb {C} by sheaves acyclic for the functor j_*.
Then
1056Equationhodge-theory-ii-3.1.7.1hodge-theory-ii-3.1.7.1.xml3.1.7.1hodge-theory-ii-3.1.7 \operatorname {H} ^ \bullet (X^*, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X^*, \Omega _{X^*}^ \bullet ) \xleftarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X,j_* \Omega _{X^*}^ \bullet )
and the Leray spectral sequence for the morphism j can be identified with the hypercohomology spectral sequence for j_* \Omega _{X^*}^ \bullet corresponding to the filtration \tau by the subcomplexes \tau _{ \leqslant -n}(j_* \Omega _{X^*}^ \bullet ) ().
1266Propositionhodge-theory-ii-3.1.8hodge-theory-ii-3.1.8.xml3.1.8hodge-theory-ii-3.1
The morphisms of filtered complexes
( \Omega _X^ \bullet \langle Y \rangle , W) \xleftarrow { \alpha } ( \Omega _X^ \bullet \langle Y \rangle , \tau ) \xhookrightarrow { \beta } (j_* \Omega _{X^*}^ \bullet , \tau )
are filtered quasi-isomorphisms.
They define an isomorphism between the Leray spectral sequence for j in complex cohomology and the hypercohomology spectral sequence of the filtered complex ( \Omega _X^ \bullet \langle Y \rangle , W) on X.
257Proofhodge-theory-ii-3.1.8
By and , it suffices to prove the first claim.
In [D1970, II, (6.9)] or [AH1955], one can find a proof of the fact that \beta is a quasi-isomorphism, and thus a filtered quasi-isomorphism.
We can also directly calculate the cohomology sheaves of the two sides: those of \Omega _X^ \bullet \langle Y \rangle are determined by [hodge-theory-ii-3.1.5.2 (?)], while those of j_* \Omega _{X^*}^ \bullet are the \mathrm {R} ^ij_* \mathbb {C}, which can be calculated by topological methods.
For n \geqslant p, we have
W_n( \Omega _X^p \langle Y \rangle ) = \Omega _X^p \langle Y \rangle
and so \alpha is a morphism from \Omega _X^ \bullet \langle Y \rangle, endowed with \tau, to \Omega _X^ \bullet \langle Y \rangle, endowed with the decreasing filtration associated to W ().
By [hodge-theory-ii-3.1.5.2 (?)], we have
256Equationhodge-theory-ii-3.1.8.1hodge-theory-ii-3.1.8.1.xml3.1.8.1hodge-theory-ii-3.1.8 \operatorname { \mathscr {H}} ^i( \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle )) = \begin {cases} 0 & \text {for }i \neq n \\ \varepsilon _X^n & \text {for }i=n \end {cases} \tag{3.1.8.1}
and we deduce from the first line of this equation that \alpha is a filtered quasi-isomorphism.
This proves .
By , the isomorphism defines an isomorphism
258Equationhodge-theory-ii-3.1.8.2hodge-theory-ii-3.1.8.2.xml3.1.8.2hodge-theory-ii-3.1.8 \mathrm {R} ^n j_* \mathbb {C} \simeq \operatorname { \mathscr {H}} ^n(j_* \Omega _{X^*}^ \bullet ) \simeq \operatorname { \mathscr {H}} ^n( \Omega _X^ \bullet \langle Y \rangle ) \simeq \varepsilon _X^n. \tag{3.1.8.2}
The isomorphisms correspond, via , to the cup product.
1268Propositionhodge-theory-ii-3.1.9hodge-theory-ii-3.1.9.xml3.1.9hodge-theory-ii-3.1
The canonical morphism from \mathrm {R} ^nj_* \mathbb {Z} to \mathrm {R} ^nj_* \mathbb {C} identifies, via , the sheaf \mathrm {R} ^nj_* \mathbb {Z} with ( \varepsilon _X^n)_ \mathbb {Z} ().
1267Proofhodge-theory-ii-3.1.9
The question is local on X.
We can thus suppose that X is an open polycylinder D^m, with
D = \{ z \in \mathbb {C} \mid |z|<1 \}
and also that Y= \bigcup _{k=1}^ \ell Y_k with Y_k= \operatorname {pr} _k^{-1}(0).
The fibre at 0 of \mathrm {R} ^nj_* \mathbb {Z} is then the integer cohomology of X^*=(D^*)^ \ell \times D^{m- \ell }, with
D^* = \{ z \in \mathbb {C} \mid 0<|z|<1 \} .
The space X^* has the homotopy type of a torus;
its cohomology is thus torsion free, and the cup product defines isomorphisms
\bigwedge ^n( \mathrm {R} ^1j_* \mathbb {Z} ) \xrightarrow { \sim } ( \mathrm {R} ^nj_* \mathbb {Z} )_0.
It thus suffices to prove for n=1.
The integer homology \operatorname {H} _1(X^*) is generated by the loops \gamma _k that go around the various Y_k.
We have
\oint _{ \gamma _k} \frac { \mathrm {d} z_k}{z_k} = \pm2 \pi i
and so the integer cohomology is generated by the \frac {1}{2 \pi i} \frac { \mathrm {d} z_k}{z_k}, and this proves .
1271hodge-theory-ii-3.1.10hodge-theory-ii-3.1.10.xml3.1.10hodge-theory-ii-3.1
Let \mathscr { F } be a coherent analytic sheaf on X^*, given as the restriction to X^* of a coherent analytic sheaf \mathscr { F } ' on X.
We define the meromorphic direct image j_*^ \mathrm {m} \mathscr { F } of \mathscr { F } to be the inductive limit
j_*^ \mathrm {m} \mathscr { F } = \varinjlim \mathscr { F } '(nY).
Locally on X, Y is the sum of a finite family (Y_i)_{i \in I} of smooth divisors, and we define the pole-order filtration P on j_*^ \mathrm {m} \mathcal {O} _X^* by the equation
1269Equationhodge-theory-ii-3.1.10.1hodge-theory-ii-3.1.10.1.xml3.1.10.1hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \mathcal {O} _{X^*}) = \sum _{n \in A_p} \mathcal {O} _X \left ( \sum (n_i+1)Y_i \right ) \tag{3.1.10.1}
where
A_p = \left \{ (n_i)_{i \in I} \mid \sum _i n_i \leqslant -p \text { and } n_i \geqslant0 \text { for all } i \right \} .
This construction globalises by endowing j_*^ \mathrm {m} \mathcal {O} _{X^*} with an exhaustive filtration such that P^p=0 for p>0.
We define the pole-order filtration of the complex j_*^ \mathrm {m} \Omega _{X^*}^ \bullet =j_*^ \mathrm {m} \mathcal {O} _{X^*} \otimes \Omega _X^ \bullet to be the filtration
1270Equationhodge-theory-ii-3.1.10.2hodge-theory-ii-3.1.10.2.xml3.1.10.2hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \Omega _{X^*}^k) = P^{p-k}(j_*^ \mathrm {m} \mathcal {O} _X) \otimes \Omega _X^k. \tag{3.1.10.2}
The filtration P induces, on the subcomplex \Omega _X^ \bullet \langle Y \rangle of j_*^ \mathrm {m} \Omega _{X^*}^ \bullet, the stupid filtration by the \sigma _{ \geqslant p}( \Omega _X^ \bullet \langle Y \rangle ), which we also call the Hodge filtration F.
1272Propositionhodge-theory-ii-3.1.11hodge-theory-ii-3.1.11.xml3.1.11hodge-theory-ii-3.1
The inclusion morphism
( \Omega _X^ \bullet \langle Y \rangle , F) \to (j_*^ \mathrm {m} \Omega _{X_*}^ \bullet , P)
is a filtered quasi-isomorphism.
254Proofhodge-theory-ii-3.1.11
This statement was suggested to me by [G1969].
A proof can be found in [D1970, II, (3.13)].
1644hodge-theory-ii-3.2hodge-theory-ii-3.2.xmlMixed Hodge theory3.2hodge-theory-ii-3
We recall that, from here on, we say "scheme" to mean a scheme of finite type over \mathbb {C}, and "sheaf on S" to mean a sheaf on S^ \mathrm {an}.
484hodge-theory-ii-3.2.1hodge-theory-ii-3.2.1.xml3.2.1hodge-theory-ii-3.2
Let X be a smooth and separated scheme.
By Nagata [N1962], X is a Zariski open of a complete scheme \bar {X}.
By Hironaka [H1964], we can take \bar {X} to be smooth and such that Y= \bar {X} \setminus X is a normal crossing divisor.
The reader who wishes to avoid the reference to Nagata can suppose X to be quasi-projective.
The smooth completion \bar {X} can then be chosen to be projective and such that Y is the union of smooth divisors.
If we limit ourselves to such compactifications, then we only need Hodge theory in its standard form ().
485hodge-theory-ii-3.2.2hodge-theory-ii-3.2.2.xml3.2.2hodge-theory-ii-3.2
By and , we have
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \simeq \operatorname { \mathbb {H}} ^ \bullet ( \bar {X}, \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ).
We define the Hodge filtration F on the complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle as the filtration F^p= \sigma _{ \geqslant p} given by the stupid truncations ().
On \Omega _{ \bar {X}}^ \bullet \langle Y \rangle, we thus have two filtrations: F and W ().
489hodge-theory-ii-3.2.3hodge-theory-ii-3.2.3.xml3.2.3hodge-theory-ii-3.2
We will need to make use of the fact that there exist bifiltered resolutions i \colon \Omega _{ \bar {X}}^ \bullet \langle Y \rangle \to K^ \bullet such that the \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j) are \Gamma-acyclic sheaves:
\operatorname {H} ^i( \bar {X}, \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j)) = 0 \qquad \text {for }i>0.
Here are two methods to construct such a resolution:
We can take K^ \bullet to be the canonical Godement resolution \mathscr {C}^ \bullet ( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), filtered by the \mathscr {C}^ \bullet (W_n( \Omega _{ \bar {X}} \langle Y \rangle )) and the \mathscr {C}^ \bullet (F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )).
This is a bifiltered resolution since \mathscr {C}^ \bullet is an exact functor.
We can take K^ \bullet to be the \mathrm {d} ''-resolution of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
Let \Omega _{ \bar {X}}^{pq} be the sheaf of C^ \infty forms of type (p,q);
then K^ \bullet is the simple complex associated to the double complex of the \Omega _{ \bar {X}}^p \langle Y \rangle \otimes _ \mathcal {O} \Omega _{ \bar {X}}^{0,q} (a subcomplex of the j_* \Omega _X^{ \bullet \bullet }).
This complex is filtered by the F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet } and by the W_n( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet };
to prove that this is a bifiltered resolution, we use the fact that the sheaf \mathcal {O} _ \infty of complex C^ \infty functions on \bar {X} is flat over \mathcal {O} (a corollary of the Malgrange C^ \infty preparation theorem).
The sheaves \operatorname {Gr} _F \operatorname {Gr} _W(K^ \bullet ) are fine, since they are sheaves of modules over the soft sheaf \mathcal {O} _ \infty.
492hodge-theory-ii-3.2.4hodge-theory-ii-3.2.4.xml3.2.4hodge-theory-ii-3.2
With the notation of , the complex cohomology of X appears as the cohomology of the bifiltered complex \Gamma ( \bar {X},K^ \bullet ).
We thus have two spectral sequences abutting to \operatorname {H} ^ \bullet (X, \mathbb {C} ).
They can be written, with the notation of , as:
490Equationhodge-theory-ii-3.2.4.1hodge-theory-ii-3.2.4.1.xml3.2.4.1hodge-theory-ii-3.2.4 {}_WE_1^{pq} = \operatorname { \mathbb {H}} ^{p+q}( \bar {X}, \varepsilon _{ \bar {X}}^{-p}[p]) = \operatorname { \mathbb {H}} ^{2p+q}( \widetilde {Y}^p, \varepsilon ^{-q}) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ) \tag{3.2.4.1}
491Equationhodge-theory-ii-3.2.4.2hodge-theory-ii-3.2.4.2.xml3.2.4.2hodge-theory-ii-3.2.4 {}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ). \tag{3.2.4.2}
The first of these, up to the renumbering {}_WE_1^{pq} \mapsto E_2^{2p+q,-p}, is exactly the Leray spectral sequence of the inclusion j.
493Theoremhodge-theory-ii-3.2.5hodge-theory-ii-3.2.5.xml3.2.5hodge-theory-ii-3.2
On the pages {}_WE_r^{pq} of the spectral sequence , the first direct filtration, the second direct filtration, and the recurrent filtration defined by F all coincide.
The filtration on \operatorname {H} ^n(X, \mathbb {C} ) that is the abutment of the spectral sequence {}_WE comes from a filtration W of \operatorname {H} ^n(X, \mathbb {Q} ).
Neither it, nor the filtration F that is the abutment of the spectral sequence {}_FE, depend on the choice of compactification \bar {X} of X or on the choice of K^ \bullet.
The filtrations W[n] () and F define on \operatorname {H} ^n(X, \mathbb {Z} ) a mixed Hodge structure, functorially in X.
By , the spectral sequence {}_WE is the Leray spectral sequence for j_* (up to renumbering).
It is thus induced by tensoring a \mathbb {Q}-vectorial spectral sequence with \mathbb {C}, and so the first claim of (ii) is true.
Complex conjugation acts on the {}_WE;
it can be calculated via .
494Lemmahodge-theory-ii-3.2.6hodge-theory-ii-3.2.6.xml3.2.6hodge-theory-ii-3.2
The hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), endowed with the filtration induced by the Hodge filtration, degenerate at the E_1 page.
369Proofhodge-theory-ii-3.2.6
Define the Y^n and the \widetilde {Y}^n as in , and let i_n \colon \widetilde {Y}^n \to X.
By [hodge-theory-ii-3.1.5.2 (?)], we have
\operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \sim (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n].
Furthermore, the Hodge filtration induces the stupid filtration (), and so the spectral sequence in is given by translations of the degrees of the classical spectral sequence
E_1^{pq} = \operatorname {H} ^q( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^p( \varepsilon ^n)) \Rightarrow \operatorname {H} ^{p+q}( \widetilde {Y}^n, \varepsilon ^n).
If \bar {X} is projective and Y the union of smooth divisors, then \widetilde {Y}^n is projective, \varepsilon ^n is a trivial local system, and classical Hodge theory () gives the degeneration claimed in .
For the general case, refer to [D1968] (see also and ).
Hodge theory also tells us that the Hodge filtration on \operatorname {H} ^k( \widetilde {Y}^n, \varepsilon ^n) is k-opposite to its complex conjugate (the complex conjugate being defined in terms of \varepsilon _ \mathbb {Z} ^n ()).
We have, taking into account the translations of the degrees:
495Lemmahodge-theory-ii-3.2.7hodge-theory-ii-3.2.7.xml3.2.7hodge-theory-ii-3.2
The filtration on
{}_WE_1^{-n,k+n} = \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \approx \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n)
which is the abutment of the spectral sequence in is (k+n)-opposite to its complex conjugate.
498Lemmahodge-theory-ii-3.2.8hodge-theory-ii-3.2.8.xml3.2.8hodge-theory-ii-3.2
The differentials d_1 of the spectral sequence {}_WE are strictly compatible with the filtration F.
497Proofhodge-theory-ii-3.2.8
On the E_1 pages, there is only the filtration induced by F to consider ( and (iii) of ), and d_1 is compatible with this filtration ((i) of ).
This filtration is the abutment of the spectral sequence in ((ii) of ).
By , the arrow
d_1 \colon \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \to \operatorname { \mathbb {H}} ^{k+1}( \bar {X}, \operatorname {Gr} _{n-1}^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ))
or
496Equationhodge-theory-ii-3.2.8.1hodge-theory-ii-3.2.8.1.xml3.2.8.1hodge-theory-ii-3.2.8 d_1 \colon \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n) \to \operatorname {H} ^{k-n+1}( \widetilde {Y}^{n-1}, \varepsilon ^{n-1}) \tag{3.2.8.1}
is compatible with the filtrations that are (k+n)-opposite to their complex conjugate.
Since d_1 commutes up to complex conjugation, d_1 respects the bigrading (of weight k+n) defined by F and \bar {F}, which proves .
Furthermore, the cohomology of the complex E_1 is again bigraded:
499Lemmahodge-theory-ii-3.2.9hodge-theory-ii-3.2.9.xml3.2.9hodge-theory-ii-3.2
The recurrent filtration F on {}_WE_2^{pq} is q-opposite to its complex conjugate.
We prove by induction on r that:
501Lemmahodge-theory-ii-3.2.10hodge-theory-ii-3.2.10.xml3.2.10hodge-theory-ii-3.2
For r \geqslant0, the differentials d_r of the spectral sequence {}_WE are strictly compatible with the recurrent filtration F.
For r \geqslant2, they are zero.
500Proofhodge-theory-ii-3.2.10
For r=0 (resp. r=1) we apply and (iii) of (resp. ).
For r \geqslant2, it suffices to prove that d_r=0.
By induction, and following , we can assume that, on the pages {}_WE^s (for s \geqslant r+1), we have F_d=F_r=F_{d^*}, and that {}_WE_r={}_WE_2.
By (i) of , d_r is thus compatible with the filtration F_r.
On {}_WE_r^{pq}={}_WE_2^{pq}, the filtration F_r is q-opposite to its complex conjugate.
The morphism
d_r \colon {}_WE_r^{pq} \to {}_WE_r^{p+r,q-r+1}
thus satisfies, for r-1>0,
\begin {aligned} d_r({}_WE_r^{pq}) &= d_r \left ( \sum _{a+b=q} \Big ( F^a({}_WE_r^{pq}) \cap \bar {F}^b({}_WE_r^{pq}) \Big ) \right ) \\ & \subset \sum _{a+b=q} \Big ( F^a({}_WE_r^{p+r,q-r+1}) \cap \bar {F}^b({}_WE_r^{p+r,q-r+1}) \Big ) \\ &= 0. \end {aligned}
This proves .
, using , proves (i) of .
By , the filtration on {}_WE_ \infty ^{pq} induced by the filtration F of \operatorname {H} ^{p+q}(X, \mathbb {C} ) is q-opposite to its complex conjugate.
Since q=-p+(p+q), this proves the first part of (iii) of 507hodge-theory-ii-3.2.11hodge-theory-ii-3.2.11.xml3.2.11hodge-theory-ii-3.2
We now prove (ii) and (iii) of , which will finish the proof.
Independence from the choice of K^ \bullet.
The filtrations F and W of \operatorname {H} ^ \bullet (X, \mathbb {C} ) are the abutments of the hypercohomology spectral sequences of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle for the filtrations F and W.
These whole spectral sequences do not depend on the choice of K^ \bullet.
Functoriality.
Let f \colon X_1 \to X_2 be a morphism of schemes.
Suppose that we are given a morphism of smooth compactifications
505hodge-theory-ii-3.2.11.1hodge-theory-ii-3.2.11.1.xml3.2.11.1hodge-theory-ii-3.2.11 \begin {CD} X_1 @>f>> X_2 \\ @V{j_1}VV @VV{j_2}V \\ \bar {X}_1 @>>{ \bar {f}}> \bar {X}_2 \end {CD} \tag{3.2.11.1}
with the Y_i= \bar {X}_i \setminus X_i normal crossing divisors.
The canonical morphism (cf. ) from \bar {f}^* \Omega _{ \bar {X}_2}^ \bullet \langle Y_2 \rangle to \Omega _{ \bar {X}_1}^ \bullet \langle Y_1 \rangle is then a morphism of bifiltered complexes;
on hypercohomology, it induces a morphism compatible with F and W, and thus f^* \colon \operatorname {H} ^n(X_2, \mathbb {Z} ) \to \operatorname {H} ^n(X_1, \mathbb {Z} ) is a morphism of mixed Hodge structures, for the structures defined by the compactifications \bar {X}_i.
Independence from the choice of compactification.
With the notation of (B), if f is an isomorphism, then f^* is a bijective morphism of mixed Hodge structures, and thus an isomorphism ().
If \bar {X}_1 and \bar {X}_2 are two smooth compactifications of X, with Y_i= \bar {X}_i \setminus X smooth crossing divisors, then there exists a third smooth compactification \bar {X}, with Y= \bar {X} \setminus X a smooth crossing divisor, that fits into a commutative diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& X \ar [dl,hook] \ar [d,hook] \ar [dr,hook]
\\ \bar {X}_1
& \bar {X} \ar [l] \ar [r]
& \bar {X}_2
\end {tikzcd}
Namely, we can take \bar {X} to be a resolution of singularities of the closure of the diagonal image of X in \bar {X}_1 \times \bar {X}_2.
The identity map from \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_1 to \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_2 is thus the composite of two isomorphisms.
To finish the proof of (ii) and (iii), we note that every morphism f fits into a diagram of the form in : we choose compactifications \bar {X}'_1 and \bar {X}_2 of X_1 and X_2, then we take \bar {X}_1 to be a resolution of singularities of the closure of the image of X_1 in \bar {X}'_1 \times \bar {X}_2.
508Definitionhodge-theory-ii-3.2.12hodge-theory-ii-3.2.12.xml3.2.12hodge-theory-ii-3.2
The mixed Hodge structure of the cohomology of a smooth separated algebraic variety is the mixed Hodge structure from (iii) of .
515Corollaryhodge-theory-ii-3.2.13hodge-theory-ii-3.2.13.xml3.2.13hodge-theory-ii-3.2
With the above notation:
The spectral sequence in degenerates at E_2, i.e. the Leray spectral sequence for the inclusion j \colon X^* \hookrightarrow X degenerates at E_3 (i.e. E_3=E_ \infty).
The spectral sequence in
{}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} )
degenerates at E_1.
The spectral sequence defined by the sheaf \Omega _X^p \langle Y \rangle, endowed with the filtration W
\begin {aligned} E_1^{-n,k+n} &= \operatorname {H} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _X^p \langle Y \rangle )) \\ & \approx \operatorname {H} ^k( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^{p-n}( \varepsilon ^n)) \Rightarrow \operatorname {H} ^k( \bar {X}, \Omega _X^p \langle Y \rangle ) \end {aligned}
degenerates at E_2.
514Proofhodge-theory-ii-3.2.13
Claim (i) is proven in .
Consider the four spectral sequences in , with respect to the bifiltered complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
By (i), we have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum _{p,q} \dim {}_WE_2^{p,q}.
The {}_WE_1^{n,k-n} pages are, for fixed n, the abutments of a spectral sequence that degenerates at E_1 () with initial pages E_1^{p,n,k-p-n} (with the notation of ) the initial pages of the spectral sequence in (iii).
Since {}_Wd_1 is strictly compatible with the filtration F which is the abutment of this spectral sequence (), and is () the abutment of a morphism between these spectral sequences, which starts with the differentials from (iii), we have
\operatorname {Gr} _F^p({}_WE_2^{n,k-n}) \approx \operatorname {H} ^ \bullet (E_1^{p,n-1,k-n-p} \to E_1^{p,n,k-n-p} \to E_1^{p,n+1,k-n-p})
and \operatorname {Gr} _F^ \bullet ({}_WE_2^{ \bullet \bullet }) is the sum of the pages E_2^{ \bullet \bullet \bullet } of the spectral sequences in (iii).
We thus have
513Equationhodge-theory-ii-3.2.13.1hodge-theory-ii-3.2.13.1.xml3.2.13.1hodge-theory-ii-3.2.13 \sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum \dim E_2^{ \bullet \bullet \bullet }. \tag{3.2.13.1}
We also have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) \leqslant \dim {}_FE_1^{ \bullet \bullet }
with equality if and only if the spectral sequence in (ii) degenerates at E_1, and
\sum \dim {}_FE_1^{ \bullet \bullet } \leqslant \sum \dim E_2^{ \bullet \bullet \bullet }
with equality if and only if the spectral sequences in (iii) degenerate at E_2.
Comparing with , we obtain .
517Corollaryhodge-theory-ii-3.2.14hodge-theory-ii-3.2.14.xml3.2.14hodge-theory-ii-3.2
Let \omega be a meromorphic differential p-form on \bar {X} that is holomorphic on X and presents at worst logarithmic poles along Y.
Then the restriction \omega |X of \omega to X is closed, and if the cohomology class in \operatorname {H} ^p(X, \mathbb {C} ) defined by \omega is zero then \omega =0.
516Proofhodge-theory-ii-3.2.14
This is the particular case of (ii) of where {}_FE_1^{p0}={}_FE_ \infty ^{p0}.
522Corollaryhodge-theory-ii-3.2.15hodge-theory-ii-3.2.15.xml3.2.15hodge-theory-ii-3.2
If X is a smooth complete algebraic variety, then the mixed Hodge structure on \operatorname {H} ^n(X, \mathbb {Z} ) is the classical Hodge structure of weight n.
The Hodge numbers h^{pq} of the mixed Hodge structure of \operatorname {H} ^n(X, \mathbb {Z} ) (with X algebraic and smooth) can only be zero for p \leqslant n, q \leqslant n, and p+q \geqslant n
521Proofhodge-theory-ii-3.2.15
Claim (i) is clear;
to prove (ii), note that, if Y is the union of smooth divisors, then the rational Hodge structure \operatorname {Gr} _k^W( \operatorname {H} ^n(X, \mathbb {Q} )) is the quotient of a sub-object of a Hodge structure of weight n+k, namely
\operatorname {H} ^{n-k}( \widetilde {Y}^n, \mathbb {Q} ) \otimes \mathbb {Q} (-k).
523hodge-theory-ii-3.2.16hodge-theory-ii-3.2.16.xml3.2.16hodge-theory-ii-3.2
Let X be a smooth separated scheme.
We know that X admits smooth compactifications \bar {X} and that the subgroup of \operatorname {H} ^n(X, \mathbb {Z} ) given by the image of \operatorname {H} ^n( \bar {X}, \mathbb {Z} ) is independent of the choice of \bar {X} (cf. [G1968, (9.1) to (9.4)]).
525Corollaryhodge-theory-ii-3.2.17hodge-theory-ii-3.2.17.xml3.2.17hodge-theory-ii-3.2
Under the hypotheses of , the image of \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) in \operatorname {H} ^n(X, \mathbb {Q} ) is W_n( \operatorname {H} ^n(X, \mathbb {Q} )) (where W denotes the weight filtration ()).
524Proofhodge-theory-ii-3.2.17
We can suppose that \bar {X} \setminus X is a normal crossing divisor.
The claim then follows from the fact that W[-n] is the abutment of the Leray spectral sequence for the inclusion j \colon X \hookrightarrow \bar {X}.
527Corollaryhodge-theory-ii-3.2.18hodge-theory-ii-3.2.18.xml3.2.18hodge-theory-ii-3.2
Let f be a morphism from a smooth proper scheme Y to a smooth scheme X, with X admitting a smooth compactification j \colon X \hookrightarrow \bar {X}.
Then the groups \operatorname {H} ^n(X, \mathbb {Q} ) and \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) have the same image in \operatorname {H} ^n(Y, \mathbb {Q} ).
526Proofhodge-theory-ii-3.2.18
Since f^* and (jf)^* are strictly compatible with the weight filtration ((iii) of ), it suffices to prove that \operatorname {Gr} ^W(f^*) and \operatorname {Gr} ^W(f^*j^*) have the same image in \operatorname {Gr} ^W( \operatorname {H} ^n(Y, \mathbb {Q} )).
By and , \operatorname {Gr} _n^W(j^*) is an isomorphism, while \operatorname {Gr} _m^W(f^*)=0 for m \neq n since \operatorname {Gr} _m^W( \operatorname {H} ^n(Y, \mathbb {Q} ))=0 for m \neq n.
528Remarkhodge-theory-ii-3.2.19hodge-theory-ii-3.2.19.xml3.2.19hodge-theory-ii-3.2
By and , under the hypotheses of , the Hodge filtration on \operatorname {H} ^n(X, \mathbb {C} ) is the abutment of the hypercohomology spectral sequence of the complex j_*^m \Omega _{X^*}^ \bullet endowed with the filtration given by the pole-order filtration (): this spectral sequence coincides with .
1697hodge-theory-ii-2hodge-theory-ii-2.xmlHodge Theory II › Hodge structures2hodge-theory-ii1639hodge-theory-ii-2.1hodge-theory-ii-2.1.xmlPure structures2.1hodge-theory-ii-2773hodge-theory-ii-2.1.1hodge-theory-ii-2.1.1.xml2.1.1hodge-theory-ii-2.1
In all the following, we denote by \mathbb {C} an algebraic closure of \mathbb {R}, and we do not suppose to have chosen a root i of the equation x^2+1=0.
The theory will be invariant under complex conjugation (cf. [hodge-theory-ii-2.1.14 (?)]).
774hodge-theory-ii-2.1.2hodge-theory-ii-2.1.2.xml2.1.2hodge-theory-ii-2.1
We denote by S the real algebraic group \mathbb {C} ^*, given by restriction of scalars à la Weil from \mathbb {C} to \mathbb {R} of the group \mathbb {G}_ \mathrm {m}:
\begin {aligned} S &= \prod _{ \mathbb {C} {/} \mathbb {R} } \mathbb {G}_ \mathrm {m} \\ S( \mathbb {R} ) &= \mathbb {C} ^*. \end {aligned}
The group S is a torus, i.e. it is connected and of multiplicative type.
It is thus described by the free abelian group of finite type
X(S) = \operatorname {Hom} (S_ \mathbb {C} , \mathbb {G}_ \mathrm {m} ) = \underline { \operatorname {Hom} } (S, \mathbb {G}_ \mathrm {m} )( \mathbb {C} )
of its complex characters, endowed with the action of \operatorname {Gal} ( \mathbb {C} {/} \mathbb {R} )= \mathbb {Z} {/}(2).
The group X(S) has generators z and \bar {z}, which induce (respectively) the identity and complex conjugation:
\mathbb {C} ^* = S( \mathbb {R} ) \to S( \mathbb {C} ) \to \mathbb {G}_ \mathrm {m} ( \mathbb {C} ) = \mathbb {C} ^*.
Complex conjugation exchanges z and \bar {z}.
780hodge-theory-ii-2.1.3hodge-theory-ii-2.1.3.xml2.1.3hodge-theory-ii-2.1
We have a canonical map
775Equationhodge-theory-ii-2.1.3.1hodge-theory-ii-2.1.3.1.xml2.1.3.1hodge-theory-ii-2.1.3 w \colon \mathbb {G}_ \mathrm {m} \to S \tag{2.1.3.1}
that, on the real points, induces the inclusion of \mathbb {R} ^* into \mathbb {C} ^*.
We have
776Equationhodge-theory-ii-2.1.3.2hodge-theory-ii-2.1.3.2.xml2.1.3.2hodge-theory-ii-2.1.3 zw = \bar {z}w = \mathrm {Id} . \tag{2.1.3.2}
We also have a map
777Equationhodge-theory-ii-2.1.3.3hodge-theory-ii-2.1.3.3.xml2.1.3.3hodge-theory-ii-2.1.3 N \colon S \to \mathbb {G}_ \mathrm {m} \tag{2.1.3.3}
that on the real points can be identified with the norm N_{ \mathbb {C} {/} \mathbb {R} } \colon \mathbb {C} ^* \to \mathbb {R} ^*.
We have
778Equationhodge-theory-ii-2.1.3.4hodge-theory-ii-2.1.3.4.xml2.1.3.4hodge-theory-ii-2.1.3 N = z \bar {z} \tag{2.1.3.4}
779Equationhodge-theory-ii-2.1.3.5hodge-theory-ii-2.1.3.5.xml2.1.3.5hodge-theory-ii-2.1.3 N \circ w = (x \mapsto x^2). \tag{2.1.3.5} 781Definitionhodge-theory-ii-2.1.4hodge-theory-ii-2.1.4.xml2.1.4hodge-theory-ii-2.1
A real Hodge structure is a real vector space V of finite dimension endowed with an action of the real algebraic group S.
783hodge-theory-ii-2.1.5hodge-theory-ii-2.1.5.xml2.1.5hodge-theory-ii-2.1
By the general theory of groups of multiplicative type, giving a real Hodge structure on V is equivalent to giving a bigrading V^{p,q} of V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that satisfies \overline {V^{pq}}=V^{qp}.
The action of S and the bigrading are mutually determined via the condition:
782hodge-theory-ii-2.1.5.1hodge-theory-ii-2.1.5.1.xml2.1.5.1hodge-theory-ii-2.1.5
On V^{pq}, S acts by multiplication by z^p \bar {z}^q.
784hodge-theory-ii-2.1.6hodge-theory-ii-2.1.6.xml2.1.6hodge-theory-ii-2.1
Let ( \mathbb {C} , \times ) be the multiplicative monoid, and let
\bar {S} = \prod _{ \mathbb {C} {/} \mathbb {R} } ( \mathbb {C} , \times ).
If V is a real vector space, then we can show that it is equivalent to either give an action of \bar {S} on V or to give a bigrading V^{pq} on V such that \overline {V^{pq}}=V^{qp} and V^{pq}=0 for p<0 or q<0.
785hodge-theory-ii-2.1.7hodge-theory-ii-2.1.7.xml2.1.7hodge-theory-ii-2.1
Let V be a real Hodge structure, defined by a representation \sigma of S and a bigrading V^{pq}.
The grading of V_ \mathbb {C} by the V_ \mathbb {C} ^n= \sum _{p+q=n}V^{pq} is then defined over \mathbb {R}.
We call this the weight grading.
On V^n=V \cap V_ \mathbb {C} ^n, the representation \sigma w of \mathbb {G}_ \mathrm {m} is multiplication by x^n.
We say that V is of weight n if V^{pq}=0 for p+q \neq n, i.e. if \sigma w is multiplication by x^n.
786hodge-theory-ii-2.1.8hodge-theory-ii-2.1.8.xml2.1.8hodge-theory-ii-2.1
Let V be a real Hodge structure.
The Hodge filtration on V_ \mathbb {C} is defined by
F^p(V_ \mathbb {C} ) = \sum _{p' \geqslant p} V^{p'q'}.
By , we have:
790Propositionhodge-theory-ii-2.1.9hodge-theory-ii-2.1.9.xml2.1.9hodge-theory-ii-2.1
Let n be an integer.
The construction establishes an equivalence of categories between:
the category of real Hodge structures of weight n;
the category of pairs consisting of a real vector space V of finite dimension and of a filtration F on V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that is n-opposite to its complex conjugate \bar {F}.
794Definitionhodge-theory-ii-2.1.10hodge-theory-ii-2.1.10.xml2.1.10hodge-theory-ii-2.1
A Hodge structure H, of weight n, consists of
a \mathbb {Z}-module H_ \mathbb {Z} of finite type (the "integral lattice");
a real Hodge structure of weight n on H_ \mathbb {R} = \mathbb {R} \otimes _ \mathbb {Z} H_ \mathbb {Z}.
795hodge-theory-ii-2.1.11hodge-theory-ii-2.1.11.xml2.1.11hodge-theory-ii-2.1
A morphism f \colon H \to H' is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} such that f_ \mathbb {R} \colon H_ \mathbb {R} \to H'_ \mathbb {R} is compatible with the action of S (i.e. such that f_ \mathbb {C} is compatible with the bigrading, or with the Hodge filtration).
Hodge structures of weight n form an abelian category.
If H is of weight n, and H' is of weight n', then we define a Hodge structure H \otimes H' of weight n+n' by the equations:
(H \otimes H')_ \mathbb {Z} =H_ \mathbb {Z} \otimes H'_ \mathbb {Z};
the action of S on (H \otimes H')_ \mathbb {R} =H_ \mathbb {R} \otimes H'_ \mathbb {R} is the tensor product of the actions of S on H_ \mathbb {R} and on H'_ \mathbb {R}.
The bigrading (resp. Hodge filtration) of (H \otimes H')_ \mathbb {C} =H_ \mathbb {C} \otimes H'_ \mathbb {C} is the tensor product of the bigradings (resp. Hodge filtrations (cf. )) of H_ \mathbb {C} and of H'_ \mathbb {C}.
We define in an analogous manner the Hodge structure \underline { \operatorname {Hom} } (H,H') (of weight n'-n), the Hodge structures \wedge ^p H (of weight pn), and the dual Hodge structure H^*.
The internal hom \underline { \operatorname {Hom} } above and the homomorphism group, are related by:
620Remarkhodge-theory-ii-2.1.11.1hodge-theory-ii-2.1.11.1.xml2.1.11.1hodge-theory-ii-2.1.11\operatorname {Hom} (H,H') is the subgroup of
\underline { \operatorname {Hom} } (H,H')_ \mathbb {Z} = \operatorname {Hom} _ \mathbb {Z} (H_ \mathbb {Z} ,H'_ \mathbb {Z} )
consisting of elements of degree (0,0).
The actions of S on H_ \mathbb {R}, H'_ \mathbb {R}, and \underline { \operatorname {Hom} } (H,H')_ \mathbb {R} = \operatorname {Hom} (H_ \mathbb {R} ,H'_ \mathbb {R} ) are related by
s(f(x)) = s(f)(s(x)).
This means that if f is of degree (0,0), i.e. invariant under S, then it commutes with the action of S.
796hodge-theory-ii-2.1.12hodge-theory-ii-2.1.12.xml2.1.12hodge-theory-ii-2.1
If A is a Noetherian subring of \mathbb {R}, then we define a Hodge A-structure of weight n to consist of an A-module H_A of finite type along with a real Hodge structure of weight n on H_ \mathbb {R} =H_A \otimes _A \mathbb {R}.
This definition is mostly used for A= \mathbb {Q}.
A Hodge A-structure consists of an A-module H_A of finite type along with a real Hodge structure on H_ \mathbb {R} =H_A \otimes _A \mathbb {R} such that the weight grading is defined over the field of fractions of A.
797Definitionhodge-theory-ii-2.1.13hodge-theory-ii-2.1.13.xml2.1.13hodge-theory-ii-2.1
The Tate Hodge structure \mathbb {Z} (1) is the Hodge structure of weight -2, of rank 1, of pure bidegree (-1,-1), with integral lattice 2 \pi i \mathbb {Z} \subset \mathbb {C}.
The action of S is thus given by multiplication with the inverse of the norm ().
For n \in \mathbb {Z}, we define \mathbb {Z} (n) as the n-th tensor power of \mathbb {Z} (1), so \mathbb {Z} (n) is the Hodge structure of weight -2n, of rank 1, of pure bidegree (-n,-n), with integral lattice (2 \pi i)^n \mathbb {Z} \subset \mathbb {C}.
THe action of S is multiplication by N(x)^{-n}.
798hodge-theory-ii-2.1.14hodge-theory-ii-2.1.14.xml2.1.14hodge-theory-ii-2.1
The choice in \mathbb {C} of a solution i of the equation x^2+1=0 determines, on each complex variety X of pure dimension n, an orientation \operatorname {or} _i(X).
Replace i by -i gives
\operatorname {or} _{-i}(X) = (-1)^n \operatorname {or} _i(X).
The choice of i also defines an element C of order 4 in S( \mathbb {R} ), given by the image of i under the isomorphism S( \mathbb {R} ) \simeq \mathbb {C} ^*.
Finally, it also defines an isomorphism between \mathbb {Z} and the integral lattice of \mathbb {Z} (n), given by multiplication by (2 \pi i)^n.
When i, an orientation of X, C, or an identification \mathbb {Z} \sim \mathbb {Z} (n)_ \mathbb {Z} appear in an equation, it is implicitly understood that they all follow from a single choice of the same i, and that by replacing i with -i we obtain an equivalent equation.
799Definitionhodge-theory-ii-2.1.15hodge-theory-ii-2.1.15.xml2.1.15hodge-theory-ii-2.1
A polarisation of a Hodge structure H of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {Z} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
800hodge-theory-ii-2.1.16hodge-theory-ii-2.1.16.xml2.1.16hodge-theory-ii-2.1
The real Tate Hodge structure is the real Hodge structure \mathbb {R} (1) underlying \mathbb {Z} (1).
We similarly define \mathbb {R} (n) as underlying \mathbb {Z} (1).
A polarisation of a real Hodge structure of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {R} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
A polarisation is entirely defined by the positive definite quadratic form (2 \pi i)^n(x,Cy) on H_ \mathbb {R}, imposed with only the condition of being invariant under the compact sub-torus of S given by the kernel of N.
We have
(x,y) = (Cx,Cy) = (y,C^2x) = (-1)^n(y,x).
The form (x,y) is thus symmetric or alternating, depending on the parity of n.
801hodge-theory-ii-2.1.17hodge-theory-ii-2.1.17.xml2.1.17hodge-theory-ii-2.1
The reader can generalise these definitions to Hodge A-structures of weight n ().
1640hodge-theory-ii-2.2hodge-theory-ii-2.2.xmlHodge theory2.2hodge-theory-ii-21174hodge-theory-ii-2.2.1hodge-theory-ii-2.2.1.xml2.2.1hodge-theory-ii-2.2
Let X be a compact Kähler variety (for example, smooth and projective).
By the holomorphic Poincaré lemma, the de Rham complex \Omega _X^ \bullet is a resolution of the constant sheaf \mathbb {C}.
We thus have an isomorphism ()
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \sim \operatorname { \mathbb {H}} ^ \bullet (X, \Omega _X^ \bullet )
and the stupid filtration on \Omega _X^ \bullet () defines the hypercohomology spectral sequence
388Equationhodge-theory-ii-2.2.1.1hodge-theory-ii-2.2.1.1.xml2.2.1.1hodge-theory-ii-2.2.1 E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^q) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} ) \tag{2.2.1.1}
which abuts to the Hodge filtration of \operatorname {H} ^ \bullet (X, \mathbb {C} ).
By Hodge theory [H1952; W1958], we have:
The spectral sequence degenerates: E_1=E_ \infty.
The Hodge filtration of \operatorname {H} ^n(X, \mathbb {C} ) is n-opposite to the complex conjugate filtration.
1175hodge-theory-ii-2.2.2hodge-theory-ii-2.2.2.xml2.2.2hodge-theory-ii-2.2
Let V be a local system of real vector spaces over X, i.e. a sheaf of \mathbb {R}-vector spaces that is locally isomorphic to a constant sheaf \mathbb {R} ^n.
Suppose that there exists on V a bilinear form
Q \colon V \otimes V \to \mathbb {R}
that is locally constant and defined.
If X is connected, this is the case if V is defined by a representation of a finite quotient of the fundamental group of X.
Statements (A) and (B) above remain true as they are, without needing to change the proofs, for cohomology with coefficients in V_ \mathbb {C} =V \otimes _ \mathbb {R} \mathbb {C}, since the spectral sequence
E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^p(V)) \Rightarrow \operatorname {H} ^{p+q}(X,V_ \mathbb {C} )
induced by the de Rham resolution of V_ \mathbb {C} by \Omega _X^ \bullet (V_ \mathbb {C} ) degenerates, and abuts to a filtration on \operatorname {H} ^n(X,V_ \mathbb {C} ) that is n-opposite to the complex conjugate filtration.
The vector space \operatorname {H} ^n(X,V) is thus endowed with a canonical real Hodge structure of weight n.
1176hodge-theory-ii-2.2.3hodge-theory-ii-2.2.3.xml2.2.3hodge-theory-ii-2.2
We show in [D1968] that the statements in remain true when X is a non-singular complete algebraic, not necessarily Kähler, variety.
The proof from loc. cit., based on a reduction to the projective case by Chow's lemma and resolution of singularities, extends to the case of .
1179hodge-theory-ii-2.2.4hodge-theory-ii-2.2.4.xml2.2.4hodge-theory-ii-2.2
Let \mathcal {L} be an invertible sheaf.
Here are two ways of defining the class c_1( \mathcal {L}).
1177hodge-theory-ii-2.2.4.1hodge-theory-ii-2.2.4.1.xml2.2.4.1hodge-theory-ii-2.2.4
The sheaf \mathcal {L} defines an element c in \operatorname {H} ^1(X, \mathcal {O} ^*).
Its image under \mathrm {d} f/f \colon \mathcal {O} ^* \to \Omega ^1 lies in \operatorname {H} ^1(X, \Omega ^1).
More precisely, \mathrm {d} f/f defines a morphism of complexes
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to (0 \to \Omega _X^1 \to \Omega _X^2 \to \ldots ) = \sigma _{ \geqslant1 }( \Omega _X^ \bullet ).
This complex maps to \Omega _X^ \bullet, whence
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to \Omega _X^ \bullet
and the image of c under \mathrm {d} \log is in \operatorname { \mathbb {H}} ^2( \Omega _X^ \bullet ).
This construction still makes sense for an algebraic variety over an arbitrary field k.
For k= \mathbb {C}, we further have
\operatorname {H} ^2(X, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^2(X, \Omega _X^ \bullet )
whence a class
c'_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {C} ). 1178hodge-theory-ii-2.2.4.2hodge-theory-ii-2.2.4.2.xml2.2.4.2hodge-theory-ii-2.2.4
The exponential exact sequence
0 \to \mathbb {Z} (1) \to \mathcal {O} \to \mathcal {O} ^* \to 0
defines a homomorphism
\partial \colon \operatorname {H} ^1(X, \mathcal {O} ^*) \to \operatorname {H} ^2(X, \mathbb {Z} (1))
whence a class
\partial c = c''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} (1)).
If we have a choice of i, then this class is identified with
c'''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} )
and for \mathcal {L}= \mathcal {O} (D), this c'''_1( \mathcal {L}) is exactly the integer cohomology class defined by D (with the orientations being defined by i).
1182hodge-theory-ii-2.2.5hodge-theory-ii-2.2.5.xml2.2.5hodge-theory-ii-2.2
We will prove:
1180hodge-theory-ii-2.2.5.1hodge-theory-ii-2.2.5.1.xml2.2.5.1hodge-theory-ii-2.2.5
For \alpha the natural injection of \mathbb {Z} (1) into \mathbb {C}, we have
- \alpha c''_1( \mathcal {L}) = c'_1( \mathcal {L}).
1181hodge-theory-ii-2.2.5.2hodge-theory-ii-2.2.5.2.xml2.2.5.2hodge-theory-ii-2.2.5
For \bta the natural injection of \mathbb {Z} into \mathbb {C}, we have
- \beta c'''_1( \mathcal {L}) = \frac {1}{2 \pi i}c'_1( \mathcal {L}).
We prove this by considering the following diagram of complexes, in which \approx denotes a quasi-isomorphism, and whose upper triangle anti-commutes up to homotopy:
\begin {CD} \mathbb {C} @> \approx >> \Omega _X^ \bullet @<<< \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @| @. @| \\ \mathbb {C} @<<< ( \mathbb {C} \to \mathcal {O} ) @> \approx > \mathrm {d} > \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @AAA @AAA @AA{ \mathrm {d} \log }A \\ \mathbb {Z} (1) @<<< ( \mathbb {Z} (1) \to \mathcal {O} ) @> \approx > \exp > \mathcal {O} ^*[-1] \end {CD} 1183hodge-theory-ii-2.2.6hodge-theory-ii-2.2.6.xml2.2.6hodge-theory-ii-2.2
Let X be a non-singular projective variety of pure dimension n.
A choice of i defines an orientation of X and an isomorphism \mathbb {Z} (1) \simeq \mathbb {Z}.
The corresponding trace morphism
\operatorname {H} ^{2n}(X, \mathbb {Z} (n)) \to \mathbb {Z}
that is induced does not depend on the choice of i.
By Hodge, for i \leqslant n, the morphism
L^{n-i} = - \wedge c''_1( \mathcal {O} (1))^{n-i} \colon \operatorname {H} ^i(X, \mathbb {Z} ) \to \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i))
is an isomorphism, and, combined with Poincaré duality
\operatorname {H} ^i(X, \mathbb {Z} ) \otimes \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i)) \to \operatorname {H} ^{2n}(X, \mathbb {Z} (n-i)) \to \mathbb {Z} (-i)
gives a polarisation on the primitive part \operatorname {Ker} (L^{n-i+1}) of \operatorname {H} ^i(X, \mathbb {Z} ).
We thus deduce that the rational Hodge structures \operatorname {H} ^i(X, \mathbb {Q} ) are polarisable.
1641hodge-theory-ii-2.3hodge-theory-ii-2.3.xmlMixed structures2.3hodge-theory-ii-21218Definitionhodge-theory-ii-2.3.1hodge-theory-ii-2.3.1.xml2.3.1hodge-theory-ii-2.3
A mixed Hodge structure H consists of
A \mathbb {Z}-module H_ \mathbb {Z} of finite type, called the "integral lattice";
A finite increasing filtration W_n on H_ \mathbb {Q} = \mathbb {Q} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the weight filtration;
A finite filtration F^p on H_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the Hodge filtration.
We demand that, on H_ \mathbb {C}, the filtration W_ \mathbb {C} induced by extension of scalars of W, the filtration F, and the complex conjugate \bar {F} form a system (W_ \mathbb {C} ,F, \bar {F}) of three opposite filtrations ( and ).
1219hodge-theory-ii-2.3.2hodge-theory-ii-2.3.2.xml2.3.2hodge-theory-ii-2.3
We also denote by W the filtration of H_ \mathbb {Z} given by the inverse image of the filtration W on H_ \mathbb {Q}.
The axiom of mixed Hodge structures implies that, for each n, the filtration F induces on \mathbb {C} \otimes _ \mathbb {Z} \operatorname {Gr} _n^W(H_ \mathbb {Z} ) a filtration that is n-opposite to its complex conjugate.
By , \operatorname {Gr} _n^W(H_ \mathbb {Z} ) is endowed with a Hodge structure of weight n, with Hodge filtration induced by F.
1220Examplehodge-theory-ii-2.3.3hodge-theory-ii-2.3.3.xml2.3.3hodge-theory-ii-2.3
If H is a Hodge structure of weight n, then we define a mixed Hodge structure with the same integral lattice and the same Hodge filtration by setting
W_i(H_ \mathbb {Q} ) = \begin {cases} 0 & \text {for } i<n \\ H_ \mathbb {Q} & \text {for } i \geqslant n. \end {cases} 1221hodge-theory-ii-2.3.4hodge-theory-ii-2.3.4.xml2.3.4hodge-theory-ii-2.3
A morphism f \colon H \to H' of Hodge structures is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} that is compatible with the filtrations W and F (and thus compatible with \bar {F}).
We immediately deduce from the following theorem.
1222Theoremhodge-theory-ii-2.3.5hodge-theory-ii-2.3.5.xml2.3.5hodge-theory-ii-2.3
The category of mixed Hodge structures is abelian.
The integral lattice of the kernel (resp. cokernel) of a morphism f \colon H \to H' is the kernel (resp. cokernel) K of f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z}, with K \otimes \mathbb {Q} and K \otimes \mathbb {C} being endowed with the induced filtrations (resp. quotient filtrations) of the filtrations W and F of H_ \mathbb {Q} and H_ \mathbb {C} (resp. of H'_ \mathbb {Q} and H'_ \mathbb {C}).
Every morphism f \colon H \to H' is strictly compatible with the filtration W of H_ \mathbb {Q} and H'_ \mathbb {Q} and with the filtration F of H_ \mathbb {C} and H'_ \mathbb {C}.
It induces morphisms of Hodge \mathbb {Q}-structures
\operatorname {Gr} _n^w(f) \colon \operatorname {Gr} _n^W(H_ \mathbb {Q} ) \to \operatorname {Gr} _n^W(H'_ \mathbb {Q} ).
It also induces morphisms
\operatorname {Gr} _F^p(F) \colon \operatorname {Gr} _F^p(H_ \mathbb {C} ) \to \operatorname {Gr} _F^p(H'_ \mathbb {C} )
that are strictly compatible with the filtration induced by W_ \mathbb {C}.
The functor \operatorname {Gr} _n^W is an exact functor from the category of mixed Hodge structures to the category of Hodge \mathbb {Q}-structures of weight n.
The functor \operatorname {Gr} _F^p is an exact functor.
1223hodge-theory-ii-2.3.6hodge-theory-ii-2.3.6.xml2.3.6hodge-theory-ii-2.3
Let H be a mixed Hodge structure.
The W_n(H_ \mathbb {Z} ), endowed with the filtrations induced by W and F, then form mixed Hodge substructures W_n(H) of H.
The quotient W_n(H)/W_{n-1}(H) can be identified with \operatorname {Gr} _n^W(H_ \mathbb {Z} ) endowed with its mixed Hodge structure ( and ).
We denote this Hodge structure by \operatorname {Gr} _n^W(H).
1224hodge-theory-ii-2.3.7hodge-theory-ii-2.3.7.xml2.3.7hodge-theory-ii-2.3
We set
H^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _{p+q}^W(H_ \mathbb {C} ) = ( \operatorname {Gr} _{p+q}^W(H))^{p,q}.
The Hodge numbers of H are the integers
h^{p,q} = \dim _ \mathbb {C} H^{p,q}.
The Hodge number h^{pq} of H is thus the Hodge number h^{pq} of the Hodge structure \operatorname {Gr} _{p+q}^W(H)1225hodge-theory-ii-2.3.8hodge-theory-ii-2.3.8.xml2.3.8hodge-theory-ii-2.3
We define a mixed Hodge \mathbb {Q}-structure H to consist of a vector space H_ \mathbb {Q} of finite dimension over \mathbb {Q}, a finite increasing filtration W on H_ \mathbb {Q}, and a finite decreasing filtration F on H_ \mathbb {C}, with the filtrations W_ \mathbb {C}, F, and \bar {F} being opposite.
generalises trivially to this variation.
1698hodge-theory-ii-2.2hodge-theory-ii-2.2.xmlHodge Theory II › Hodge structures › Hodge theory2.2hodge-theory-ii-21174hodge-theory-ii-2.2.1hodge-theory-ii-2.2.1.xml2.2.1hodge-theory-ii-2.2
Let X be a compact Kähler variety (for example, smooth and projective).
By the holomorphic Poincaré lemma, the de Rham complex \Omega _X^ \bullet is a resolution of the constant sheaf \mathbb {C}.
We thus have an isomorphism ()
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \sim \operatorname { \mathbb {H}} ^ \bullet (X, \Omega _X^ \bullet )
and the stupid filtration on \Omega _X^ \bullet () defines the hypercohomology spectral sequence
388Equationhodge-theory-ii-2.2.1.1hodge-theory-ii-2.2.1.1.xml2.2.1.1hodge-theory-ii-2.2.1 E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^q) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} ) \tag{2.2.1.1}
which abuts to the Hodge filtration of \operatorname {H} ^ \bullet (X, \mathbb {C} ).
By Hodge theory [H1952; W1958], we have:
The spectral sequence degenerates: E_1=E_ \infty.
The Hodge filtration of \operatorname {H} ^n(X, \mathbb {C} ) is n-opposite to the complex conjugate filtration.
1175hodge-theory-ii-2.2.2hodge-theory-ii-2.2.2.xml2.2.2hodge-theory-ii-2.2
Let V be a local system of real vector spaces over X, i.e. a sheaf of \mathbb {R}-vector spaces that is locally isomorphic to a constant sheaf \mathbb {R} ^n.
Suppose that there exists on V a bilinear form
Q \colon V \otimes V \to \mathbb {R}
that is locally constant and defined.
If X is connected, this is the case if V is defined by a representation of a finite quotient of the fundamental group of X.
Statements (A) and (B) above remain true as they are, without needing to change the proofs, for cohomology with coefficients in V_ \mathbb {C} =V \otimes _ \mathbb {R} \mathbb {C}, since the spectral sequence
E_1^{pq} = \operatorname {H} ^q(X, \Omega _X^p(V)) \Rightarrow \operatorname {H} ^{p+q}(X,V_ \mathbb {C} )
induced by the de Rham resolution of V_ \mathbb {C} by \Omega _X^ \bullet (V_ \mathbb {C} ) degenerates, and abuts to a filtration on \operatorname {H} ^n(X,V_ \mathbb {C} ) that is n-opposite to the complex conjugate filtration.
The vector space \operatorname {H} ^n(X,V) is thus endowed with a canonical real Hodge structure of weight n.
1176hodge-theory-ii-2.2.3hodge-theory-ii-2.2.3.xml2.2.3hodge-theory-ii-2.2
We show in [D1968] that the statements in remain true when X is a non-singular complete algebraic, not necessarily Kähler, variety.
The proof from loc. cit., based on a reduction to the projective case by Chow's lemma and resolution of singularities, extends to the case of .
1179hodge-theory-ii-2.2.4hodge-theory-ii-2.2.4.xml2.2.4hodge-theory-ii-2.2
Let \mathcal {L} be an invertible sheaf.
Here are two ways of defining the class c_1( \mathcal {L}).
1177hodge-theory-ii-2.2.4.1hodge-theory-ii-2.2.4.1.xml2.2.4.1hodge-theory-ii-2.2.4
The sheaf \mathcal {L} defines an element c in \operatorname {H} ^1(X, \mathcal {O} ^*).
Its image under \mathrm {d} f/f \colon \mathcal {O} ^* \to \Omega ^1 lies in \operatorname {H} ^1(X, \Omega ^1).
More precisely, \mathrm {d} f/f defines a morphism of complexes
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to (0 \to \Omega _X^1 \to \Omega _X^2 \to \ldots ) = \sigma _{ \geqslant1 }( \Omega _X^ \bullet ).
This complex maps to \Omega _X^ \bullet, whence
\mathrm {d} \log \colon \mathcal {O} ^*[-1] \to \Omega _X^ \bullet
and the image of c under \mathrm {d} \log is in \operatorname { \mathbb {H}} ^2( \Omega _X^ \bullet ).
This construction still makes sense for an algebraic variety over an arbitrary field k.
For k= \mathbb {C}, we further have
\operatorname {H} ^2(X, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^2(X, \Omega _X^ \bullet )
whence a class
c'_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {C} ). 1178hodge-theory-ii-2.2.4.2hodge-theory-ii-2.2.4.2.xml2.2.4.2hodge-theory-ii-2.2.4
The exponential exact sequence
0 \to \mathbb {Z} (1) \to \mathcal {O} \to \mathcal {O} ^* \to 0
defines a homomorphism
\partial \colon \operatorname {H} ^1(X, \mathcal {O} ^*) \to \operatorname {H} ^2(X, \mathbb {Z} (1))
whence a class
\partial c = c''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} (1)).
If we have a choice of i, then this class is identified with
c'''_1( \mathcal {L}) \in \operatorname {H} ^2(X, \mathbb {Z} )
and for \mathcal {L}= \mathcal {O} (D), this c'''_1( \mathcal {L}) is exactly the integer cohomology class defined by D (with the orientations being defined by i).
1182hodge-theory-ii-2.2.5hodge-theory-ii-2.2.5.xml2.2.5hodge-theory-ii-2.2
We will prove:
1180hodge-theory-ii-2.2.5.1hodge-theory-ii-2.2.5.1.xml2.2.5.1hodge-theory-ii-2.2.5
For \alpha the natural injection of \mathbb {Z} (1) into \mathbb {C}, we have
- \alpha c''_1( \mathcal {L}) = c'_1( \mathcal {L}).
1181hodge-theory-ii-2.2.5.2hodge-theory-ii-2.2.5.2.xml2.2.5.2hodge-theory-ii-2.2.5
For \bta the natural injection of \mathbb {Z} into \mathbb {C}, we have
- \beta c'''_1( \mathcal {L}) = \frac {1}{2 \pi i}c'_1( \mathcal {L}).
We prove this by considering the following diagram of complexes, in which \approx denotes a quasi-isomorphism, and whose upper triangle anti-commutes up to homotopy:
\begin {CD} \mathbb {C} @> \approx >> \Omega _X^ \bullet @<<< \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @| @. @| \\ \mathbb {C} @<<< ( \mathbb {C} \to \mathcal {O} ) @> \approx > \mathrm {d} > \sigma _{ \geqslant1 }( \Omega _X^ \bullet ) \\ @AAA @AAA @AA{ \mathrm {d} \log }A \\ \mathbb {Z} (1) @<<< ( \mathbb {Z} (1) \to \mathcal {O} ) @> \approx > \exp > \mathcal {O} ^*[-1] \end {CD} 1183hodge-theory-ii-2.2.6hodge-theory-ii-2.2.6.xml2.2.6hodge-theory-ii-2.2
Let X be a non-singular projective variety of pure dimension n.
A choice of i defines an orientation of X and an isomorphism \mathbb {Z} (1) \simeq \mathbb {Z}.
The corresponding trace morphism
\operatorname {H} ^{2n}(X, \mathbb {Z} (n)) \to \mathbb {Z}
that is induced does not depend on the choice of i.
By Hodge, for i \leqslant n, the morphism
L^{n-i} = - \wedge c''_1( \mathcal {O} (1))^{n-i} \colon \operatorname {H} ^i(X, \mathbb {Z} ) \to \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i))
is an isomorphism, and, combined with Poincaré duality
\operatorname {H} ^i(X, \mathbb {Z} ) \otimes \operatorname {H} ^{2n-i}(X, \mathbb {Z} (n-i)) \to \operatorname {H} ^{2n}(X, \mathbb {Z} (n-i)) \to \mathbb {Z} (-i)
gives a polarisation on the primitive part \operatorname {Ker} (L^{n-i+1}) of \operatorname {H} ^i(X, \mathbb {Z} ).
We thus deduce that the rational Hodge structures \operatorname {H} ^i(X, \mathbb {Q} ) are polarisable.
1699hodge-theory-ii-3hodge-theory-ii-3.xmlHodge Theory II › Hodge theory of non-singular algebraic varieties3hodge-theory-ii1643hodge-theory-ii-3.1hodge-theory-ii-3.1.xmlLogarithmic poles and residues3.1hodge-theory-ii-31254hodge-theory-ii-3.1.1hodge-theory-ii-3.1.1.xml3.1.1hodge-theory-ii-3.1
We recall several classical properties of "logarithmic poles" of holomorphic differential forms.
The reader will find proofs in [D1970, II, (3.1) to (3.7)], for example.
1255hodge-theory-ii-3.1.2hodge-theory-ii-3.1.2.xml3.1.2hodge-theory-ii-3.1
A divisor Y in a smooth complex analytic variety X is said to be of normal crossing if the inclusion of Y into X is locally isomorphic to the inclusion of a union of coordinate hyperplanes in \mathbb {C} ^n;
this does not imply that Y is the union of smooth divisors.
Let Y be a normal crossing divisor in X, and j the inclusion of X^*=X \setminus Y into X.
We denote by \Omega _X^1 \langle Y \rangle the locally free sub-\mathcal {O}-module of j_* \Omega _{X^*}^1 generated by \Omega _X^1 and the \mathrm {d} z_i/z_i for z_i local coordinates of a local irreducible component of Y.
The sheaf \Omega _X^p \langle Y \rangle of differential p-forms on X with logarithmic poles along Y is by the definition the locally free sub-sheaf \wedge ^p \Omega _X^1 \langle Y \rangle of j_* \Omega _{X^*}^p.
1259Propositionhodge-theory-ii-3.1.3hodge-theory-ii-3.1.3.xml3.1.3hodge-theory-ii-3.1
A section \alpha of j_* \Omega _{X^*}^p belongs to \Omega _X^p \langle Y \rangle if and only if \alpha and \mathrm {d} \alpha have at worst simple poles along the divisor Y.
The \Omega _X^p \langle Y \rangle form the smallest subcomplex of j_* \Omega _{X^*}^ \bullet that is stable under exterior product and that contains \Omega _X^ \bullet and the logarithmic differential \mathrm {d} f/f of every local section of j_* \Omega _{X^*}^p meromorphic along Y.
We call \Omega _X^ \bullet \langle Y \rangle the logarithmic de Rham complex of X along Y.
By (ii) of , this complex is contravariant in the pair (X,X^*).
1261hodge-theory-ii-3.1.4hodge-theory-ii-3.1.4.xml3.1.4hodge-theory-ii-3.1
Locally on X, Y is the union of various smooth divisors Y_i, and we denote by Y^n (resp. \widetilde {Y}^n) the union (resp. disjoint sum) of the n-fold intersections of the Y_i.
The Y^n glue to give a subspace Y^n of X, and the \widetilde {Y}^n glue to give the normalisation variety of Y^n.
We have \widetilde {Y}^0=Y^0=X, and we set \widetilde {Y}= \widetilde {Y}^1.
We define the two-element set of orientations of an n-element set E as the set of generators of \wedge ^n \mathbb {Z} ^E.
For n \geqslant2, this set is that of the conjugation classes under the alternating group of total orders on E.
If, to each point y of \widetilde {Y}^n, we associate the set of n local components of Y that contain the image in X of a neighbourhood of y in \widetilde {Y}^n, then we define on \widetilde {Y}^n a local system E_n of n-element sets.
The local system of orientations of these sets is a \mathbb {Z} {/}(2)-torsor.
This torsor defines, via the inclusion of \mathbb {Z} {/}(2) into \mathbb {C} ^*, a complex local system \varepsilon ^n of rank 1 on \widetilde {Y}^n, endowed with an isomorphism ( \varepsilon ^n)^{ \otimes2 } \simeq \mathbb {C}.
We have
\varepsilon ^n \simeq \bigwedge ^n \mathbb {C} ^{E_n}.
Locally on \widetilde {Y}^n, \varepsilon ^n is endowed with two opposite isomorphisms \pm \alpha \colon \varepsilon ^n \xrightarrow { \sim } \mathbb {C}.
We set
\varepsilon _ \mathbb {Z} ^n = \alpha ^{-1}((2 \pi i)^{-n} \mathbb {Z} ).
There is a way of seeing \varepsilon ^n, endowed with \varepsilon _ \mathbb {Z} ^n, as a twisted version of \mathbb {Z} (-n) over \widetilde {Y}^n.
We denote by \varepsilon _X^n (resp. by ( \varepsilon _X^n)_ \mathbb {Z}) the direct image of \varepsilon ^n (resp. of \varepsilon _ \mathbb {Z} ^n) under the mat from \widetilde {Y}^n to X.
We have
1260Equationhodge-theory-ii-3.1.4.1hodge-theory-ii-3.1.4.1.xml3.1.4.1hodge-theory-ii-3.1.4 \varepsilon _X^n \simeq \bigwedge ^n \varepsilon _X^1 \qquad \text {for }n \geqslant0 . \tag{3.1.4.1}
If Y is the union of distinct smooth divisors (Y_i)_{i \in I}, then the choice of a total order on I trivialises the \varepsilon ^n.
1263hodge-theory-ii-3.1.5hodge-theory-ii-3.1.5.xml3.1.5hodge-theory-ii-3.1
We denote by W_n( \Omega _X^p \langle Y \rangle ) the sub-module of \Omega _X^p \langle Y \rangle consisting of linear combinations of products
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \qquad \text {for }m \leqslant n
with \alpha holomorphic and the t_{i(j)} local coordinates of the distinct local components Y_j of Y.
We define the weight filtration of \Omega _X^ \bullet \langle Y \rangle to be the increasing filtration by the subcomplexes W_n( \Omega _X^ \bullet \langle Y \rangle ).
We have
1262Equationhodge-theory-ii-3.1.5.1hodge-theory-ii-3.1.5.1.xml3.1.5.1hodge-theory-ii-3.1.5 W_n( \Omega _X^p \langle Y \rangle ) \wedge W_m( \Omega _X^p \langle Y \rangle ) \subset W_{n+m}( \Omega _X^{p+q} \langle Y \rangle ). \tag{3.1.5.1}
If we denote by i_n the map from \widetilde {Y}^n to X, then we can show that the mapping
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \longmapsto ( \alpha |Y_{i(1)} \cap \ldots \cap Y_{i(n)}) \otimes ( \text {orientation }i(1) \ldots i(n))
defines an isomorphism of complexes
\operatorname {Res} \colon \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle ) \approx (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n]
(the Poincaré residue).
1264hodge-theory-ii-3.1.6hodge-theory-ii-3.1.6.xml3.1.6hodge-theory-ii-3.1
The interpretation that follows, in terms of [hodge-theory-ii-3.1.5.2 (?)], of the Leray spectral sequence for the inclusion of X^* into X, was pointed out to me by N. Katz.
It will allow us to prove a point that I had initially considered as evident (the first part of (ii) of [hodge-theory-ii-3.2.5 (?)]).
1265hodge-theory-ii-3.1.7hodge-theory-ii-3.1.7.xml3.1.7hodge-theory-ii-3.1
Every point of X admits a fundamental system of Stein open neighbourhoods whose intersections on X^* are again Stein.
For a coherent analytic sheaf \mathscr { F } on X^*, we thus have \mathrm {R} ^i j_* \mathscr { F } =0 for i>0.
The de Rham complex \Omega _{X^*}^ \bullet is thus a resolution of the constant sheaf \mathbb {C} by sheaves acyclic for the functor j_*.
Then
1056Equationhodge-theory-ii-3.1.7.1hodge-theory-ii-3.1.7.1.xml3.1.7.1hodge-theory-ii-3.1.7 \operatorname {H} ^ \bullet (X^*, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X^*, \Omega _{X^*}^ \bullet ) \xleftarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X,j_* \Omega _{X^*}^ \bullet )
and the Leray spectral sequence for the morphism j can be identified with the hypercohomology spectral sequence for j_* \Omega _{X^*}^ \bullet corresponding to the filtration \tau by the subcomplexes \tau _{ \leqslant -n}(j_* \Omega _{X^*}^ \bullet ) ().
1266Propositionhodge-theory-ii-3.1.8hodge-theory-ii-3.1.8.xml3.1.8hodge-theory-ii-3.1
The morphisms of filtered complexes
( \Omega _X^ \bullet \langle Y \rangle , W) \xleftarrow { \alpha } ( \Omega _X^ \bullet \langle Y \rangle , \tau ) \xhookrightarrow { \beta } (j_* \Omega _{X^*}^ \bullet , \tau )
are filtered quasi-isomorphisms.
They define an isomorphism between the Leray spectral sequence for j in complex cohomology and the hypercohomology spectral sequence of the filtered complex ( \Omega _X^ \bullet \langle Y \rangle , W) on X.
257Proofhodge-theory-ii-3.1.8
By and , it suffices to prove the first claim.
In [D1970, II, (6.9)] or [AH1955], one can find a proof of the fact that \beta is a quasi-isomorphism, and thus a filtered quasi-isomorphism.
We can also directly calculate the cohomology sheaves of the two sides: those of \Omega _X^ \bullet \langle Y \rangle are determined by [hodge-theory-ii-3.1.5.2 (?)], while those of j_* \Omega _{X^*}^ \bullet are the \mathrm {R} ^ij_* \mathbb {C}, which can be calculated by topological methods.
For n \geqslant p, we have
W_n( \Omega _X^p \langle Y \rangle ) = \Omega _X^p \langle Y \rangle
and so \alpha is a morphism from \Omega _X^ \bullet \langle Y \rangle, endowed with \tau, to \Omega _X^ \bullet \langle Y \rangle, endowed with the decreasing filtration associated to W ().
By [hodge-theory-ii-3.1.5.2 (?)], we have
256Equationhodge-theory-ii-3.1.8.1hodge-theory-ii-3.1.8.1.xml3.1.8.1hodge-theory-ii-3.1.8 \operatorname { \mathscr {H}} ^i( \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle )) = \begin {cases} 0 & \text {for }i \neq n \\ \varepsilon _X^n & \text {for }i=n \end {cases} \tag{3.1.8.1}
and we deduce from the first line of this equation that \alpha is a filtered quasi-isomorphism.
This proves .
By , the isomorphism defines an isomorphism
258Equationhodge-theory-ii-3.1.8.2hodge-theory-ii-3.1.8.2.xml3.1.8.2hodge-theory-ii-3.1.8 \mathrm {R} ^n j_* \mathbb {C} \simeq \operatorname { \mathscr {H}} ^n(j_* \Omega _{X^*}^ \bullet ) \simeq \operatorname { \mathscr {H}} ^n( \Omega _X^ \bullet \langle Y \rangle ) \simeq \varepsilon _X^n. \tag{3.1.8.2}
The isomorphisms correspond, via , to the cup product.
1268Propositionhodge-theory-ii-3.1.9hodge-theory-ii-3.1.9.xml3.1.9hodge-theory-ii-3.1
The canonical morphism from \mathrm {R} ^nj_* \mathbb {Z} to \mathrm {R} ^nj_* \mathbb {C} identifies, via , the sheaf \mathrm {R} ^nj_* \mathbb {Z} with ( \varepsilon _X^n)_ \mathbb {Z} ().
1267Proofhodge-theory-ii-3.1.9
The question is local on X.
We can thus suppose that X is an open polycylinder D^m, with
D = \{ z \in \mathbb {C} \mid |z|<1 \}
and also that Y= \bigcup _{k=1}^ \ell Y_k with Y_k= \operatorname {pr} _k^{-1}(0).
The fibre at 0 of \mathrm {R} ^nj_* \mathbb {Z} is then the integer cohomology of X^*=(D^*)^ \ell \times D^{m- \ell }, with
D^* = \{ z \in \mathbb {C} \mid 0<|z|<1 \} .
The space X^* has the homotopy type of a torus;
its cohomology is thus torsion free, and the cup product defines isomorphisms
\bigwedge ^n( \mathrm {R} ^1j_* \mathbb {Z} ) \xrightarrow { \sim } ( \mathrm {R} ^nj_* \mathbb {Z} )_0.
It thus suffices to prove for n=1.
The integer homology \operatorname {H} _1(X^*) is generated by the loops \gamma _k that go around the various Y_k.
We have
\oint _{ \gamma _k} \frac { \mathrm {d} z_k}{z_k} = \pm2 \pi i
and so the integer cohomology is generated by the \frac {1}{2 \pi i} \frac { \mathrm {d} z_k}{z_k}, and this proves .
1271hodge-theory-ii-3.1.10hodge-theory-ii-3.1.10.xml3.1.10hodge-theory-ii-3.1
Let \mathscr { F } be a coherent analytic sheaf on X^*, given as the restriction to X^* of a coherent analytic sheaf \mathscr { F } ' on X.
We define the meromorphic direct image j_*^ \mathrm {m} \mathscr { F } of \mathscr { F } to be the inductive limit
j_*^ \mathrm {m} \mathscr { F } = \varinjlim \mathscr { F } '(nY).
Locally on X, Y is the sum of a finite family (Y_i)_{i \in I} of smooth divisors, and we define the pole-order filtration P on j_*^ \mathrm {m} \mathcal {O} _X^* by the equation
1269Equationhodge-theory-ii-3.1.10.1hodge-theory-ii-3.1.10.1.xml3.1.10.1hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \mathcal {O} _{X^*}) = \sum _{n \in A_p} \mathcal {O} _X \left ( \sum (n_i+1)Y_i \right ) \tag{3.1.10.1}
where
A_p = \left \{ (n_i)_{i \in I} \mid \sum _i n_i \leqslant -p \text { and } n_i \geqslant0 \text { for all } i \right \} .
This construction globalises by endowing j_*^ \mathrm {m} \mathcal {O} _{X^*} with an exhaustive filtration such that P^p=0 for p>0.
We define the pole-order filtration of the complex j_*^ \mathrm {m} \Omega _{X^*}^ \bullet =j_*^ \mathrm {m} \mathcal {O} _{X^*} \otimes \Omega _X^ \bullet to be the filtration
1270Equationhodge-theory-ii-3.1.10.2hodge-theory-ii-3.1.10.2.xml3.1.10.2hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \Omega _{X^*}^k) = P^{p-k}(j_*^ \mathrm {m} \mathcal {O} _X) \otimes \Omega _X^k. \tag{3.1.10.2}
The filtration P induces, on the subcomplex \Omega _X^ \bullet \langle Y \rangle of j_*^ \mathrm {m} \Omega _{X^*}^ \bullet, the stupid filtration by the \sigma _{ \geqslant p}( \Omega _X^ \bullet \langle Y \rangle ), which we also call the Hodge filtration F.
1272Propositionhodge-theory-ii-3.1.11hodge-theory-ii-3.1.11.xml3.1.11hodge-theory-ii-3.1
The inclusion morphism
( \Omega _X^ \bullet \langle Y \rangle , F) \to (j_*^ \mathrm {m} \Omega _{X_*}^ \bullet , P)
is a filtered quasi-isomorphism.
254Proofhodge-theory-ii-3.1.11
This statement was suggested to me by [G1969].
A proof can be found in [D1970, II, (3.13)].
1644hodge-theory-ii-3.2hodge-theory-ii-3.2.xmlMixed Hodge theory3.2hodge-theory-ii-3
We recall that, from here on, we say "scheme" to mean a scheme of finite type over \mathbb {C}, and "sheaf on S" to mean a sheaf on S^ \mathrm {an}.
484hodge-theory-ii-3.2.1hodge-theory-ii-3.2.1.xml3.2.1hodge-theory-ii-3.2
Let X be a smooth and separated scheme.
By Nagata [N1962], X is a Zariski open of a complete scheme \bar {X}.
By Hironaka [H1964], we can take \bar {X} to be smooth and such that Y= \bar {X} \setminus X is a normal crossing divisor.
The reader who wishes to avoid the reference to Nagata can suppose X to be quasi-projective.
The smooth completion \bar {X} can then be chosen to be projective and such that Y is the union of smooth divisors.
If we limit ourselves to such compactifications, then we only need Hodge theory in its standard form ().
485hodge-theory-ii-3.2.2hodge-theory-ii-3.2.2.xml3.2.2hodge-theory-ii-3.2
By and , we have
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \simeq \operatorname { \mathbb {H}} ^ \bullet ( \bar {X}, \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ).
We define the Hodge filtration F on the complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle as the filtration F^p= \sigma _{ \geqslant p} given by the stupid truncations ().
On \Omega _{ \bar {X}}^ \bullet \langle Y \rangle, we thus have two filtrations: F and W ().
489hodge-theory-ii-3.2.3hodge-theory-ii-3.2.3.xml3.2.3hodge-theory-ii-3.2
We will need to make use of the fact that there exist bifiltered resolutions i \colon \Omega _{ \bar {X}}^ \bullet \langle Y \rangle \to K^ \bullet such that the \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j) are \Gamma-acyclic sheaves:
\operatorname {H} ^i( \bar {X}, \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j)) = 0 \qquad \text {for }i>0.
Here are two methods to construct such a resolution:
We can take K^ \bullet to be the canonical Godement resolution \mathscr {C}^ \bullet ( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), filtered by the \mathscr {C}^ \bullet (W_n( \Omega _{ \bar {X}} \langle Y \rangle )) and the \mathscr {C}^ \bullet (F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )).
This is a bifiltered resolution since \mathscr {C}^ \bullet is an exact functor.
We can take K^ \bullet to be the \mathrm {d} ''-resolution of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
Let \Omega _{ \bar {X}}^{pq} be the sheaf of C^ \infty forms of type (p,q);
then K^ \bullet is the simple complex associated to the double complex of the \Omega _{ \bar {X}}^p \langle Y \rangle \otimes _ \mathcal {O} \Omega _{ \bar {X}}^{0,q} (a subcomplex of the j_* \Omega _X^{ \bullet \bullet }).
This complex is filtered by the F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet } and by the W_n( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet };
to prove that this is a bifiltered resolution, we use the fact that the sheaf \mathcal {O} _ \infty of complex C^ \infty functions on \bar {X} is flat over \mathcal {O} (a corollary of the Malgrange C^ \infty preparation theorem).
The sheaves \operatorname {Gr} _F \operatorname {Gr} _W(K^ \bullet ) are fine, since they are sheaves of modules over the soft sheaf \mathcal {O} _ \infty.
492hodge-theory-ii-3.2.4hodge-theory-ii-3.2.4.xml3.2.4hodge-theory-ii-3.2
With the notation of , the complex cohomology of X appears as the cohomology of the bifiltered complex \Gamma ( \bar {X},K^ \bullet ).
We thus have two spectral sequences abutting to \operatorname {H} ^ \bullet (X, \mathbb {C} ).
They can be written, with the notation of , as:
490Equationhodge-theory-ii-3.2.4.1hodge-theory-ii-3.2.4.1.xml3.2.4.1hodge-theory-ii-3.2.4 {}_WE_1^{pq} = \operatorname { \mathbb {H}} ^{p+q}( \bar {X}, \varepsilon _{ \bar {X}}^{-p}[p]) = \operatorname { \mathbb {H}} ^{2p+q}( \widetilde {Y}^p, \varepsilon ^{-q}) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ) \tag{3.2.4.1}
491Equationhodge-theory-ii-3.2.4.2hodge-theory-ii-3.2.4.2.xml3.2.4.2hodge-theory-ii-3.2.4 {}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ). \tag{3.2.4.2}
The first of these, up to the renumbering {}_WE_1^{pq} \mapsto E_2^{2p+q,-p}, is exactly the Leray spectral sequence of the inclusion j.
493Theoremhodge-theory-ii-3.2.5hodge-theory-ii-3.2.5.xml3.2.5hodge-theory-ii-3.2
On the pages {}_WE_r^{pq} of the spectral sequence , the first direct filtration, the second direct filtration, and the recurrent filtration defined by F all coincide.
The filtration on \operatorname {H} ^n(X, \mathbb {C} ) that is the abutment of the spectral sequence {}_WE comes from a filtration W of \operatorname {H} ^n(X, \mathbb {Q} ).
Neither it, nor the filtration F that is the abutment of the spectral sequence {}_FE, depend on the choice of compactification \bar {X} of X or on the choice of K^ \bullet.
The filtrations W[n] () and F define on \operatorname {H} ^n(X, \mathbb {Z} ) a mixed Hodge structure, functorially in X.
By , the spectral sequence {}_WE is the Leray spectral sequence for j_* (up to renumbering).
It is thus induced by tensoring a \mathbb {Q}-vectorial spectral sequence with \mathbb {C}, and so the first claim of (ii) is true.
Complex conjugation acts on the {}_WE;
it can be calculated via .
494Lemmahodge-theory-ii-3.2.6hodge-theory-ii-3.2.6.xml3.2.6hodge-theory-ii-3.2
The hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), endowed with the filtration induced by the Hodge filtration, degenerate at the E_1 page.
369Proofhodge-theory-ii-3.2.6
Define the Y^n and the \widetilde {Y}^n as in , and let i_n \colon \widetilde {Y}^n \to X.
By [hodge-theory-ii-3.1.5.2 (?)], we have
\operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \sim (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n].
Furthermore, the Hodge filtration induces the stupid filtration (), and so the spectral sequence in is given by translations of the degrees of the classical spectral sequence
E_1^{pq} = \operatorname {H} ^q( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^p( \varepsilon ^n)) \Rightarrow \operatorname {H} ^{p+q}( \widetilde {Y}^n, \varepsilon ^n).
If \bar {X} is projective and Y the union of smooth divisors, then \widetilde {Y}^n is projective, \varepsilon ^n is a trivial local system, and classical Hodge theory () gives the degeneration claimed in .
For the general case, refer to [D1968] (see also and ).
Hodge theory also tells us that the Hodge filtration on \operatorname {H} ^k( \widetilde {Y}^n, \varepsilon ^n) is k-opposite to its complex conjugate (the complex conjugate being defined in terms of \varepsilon _ \mathbb {Z} ^n ()).
We have, taking into account the translations of the degrees:
495Lemmahodge-theory-ii-3.2.7hodge-theory-ii-3.2.7.xml3.2.7hodge-theory-ii-3.2
The filtration on
{}_WE_1^{-n,k+n} = \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \approx \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n)
which is the abutment of the spectral sequence in is (k+n)-opposite to its complex conjugate.
498Lemmahodge-theory-ii-3.2.8hodge-theory-ii-3.2.8.xml3.2.8hodge-theory-ii-3.2
The differentials d_1 of the spectral sequence {}_WE are strictly compatible with the filtration F.
497Proofhodge-theory-ii-3.2.8
On the E_1 pages, there is only the filtration induced by F to consider ( and (iii) of ), and d_1 is compatible with this filtration ((i) of ).
This filtration is the abutment of the spectral sequence in ((ii) of ).
By , the arrow
d_1 \colon \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \to \operatorname { \mathbb {H}} ^{k+1}( \bar {X}, \operatorname {Gr} _{n-1}^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ))
or
496Equationhodge-theory-ii-3.2.8.1hodge-theory-ii-3.2.8.1.xml3.2.8.1hodge-theory-ii-3.2.8 d_1 \colon \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n) \to \operatorname {H} ^{k-n+1}( \widetilde {Y}^{n-1}, \varepsilon ^{n-1}) \tag{3.2.8.1}
is compatible with the filtrations that are (k+n)-opposite to their complex conjugate.
Since d_1 commutes up to complex conjugation, d_1 respects the bigrading (of weight k+n) defined by F and \bar {F}, which proves .
Furthermore, the cohomology of the complex E_1 is again bigraded:
499Lemmahodge-theory-ii-3.2.9hodge-theory-ii-3.2.9.xml3.2.9hodge-theory-ii-3.2
The recurrent filtration F on {}_WE_2^{pq} is q-opposite to its complex conjugate.
We prove by induction on r that:
501Lemmahodge-theory-ii-3.2.10hodge-theory-ii-3.2.10.xml3.2.10hodge-theory-ii-3.2
For r \geqslant0, the differentials d_r of the spectral sequence {}_WE are strictly compatible with the recurrent filtration F.
For r \geqslant2, they are zero.
500Proofhodge-theory-ii-3.2.10
For r=0 (resp. r=1) we apply and (iii) of (resp. ).
For r \geqslant2, it suffices to prove that d_r=0.
By induction, and following , we can assume that, on the pages {}_WE^s (for s \geqslant r+1), we have F_d=F_r=F_{d^*}, and that {}_WE_r={}_WE_2.
By (i) of , d_r is thus compatible with the filtration F_r.
On {}_WE_r^{pq}={}_WE_2^{pq}, the filtration F_r is q-opposite to its complex conjugate.
The morphism
d_r \colon {}_WE_r^{pq} \to {}_WE_r^{p+r,q-r+1}
thus satisfies, for r-1>0,
\begin {aligned} d_r({}_WE_r^{pq}) &= d_r \left ( \sum _{a+b=q} \Big ( F^a({}_WE_r^{pq}) \cap \bar {F}^b({}_WE_r^{pq}) \Big ) \right ) \\ & \subset \sum _{a+b=q} \Big ( F^a({}_WE_r^{p+r,q-r+1}) \cap \bar {F}^b({}_WE_r^{p+r,q-r+1}) \Big ) \\ &= 0. \end {aligned}
This proves .
, using , proves (i) of .
By , the filtration on {}_WE_ \infty ^{pq} induced by the filtration F of \operatorname {H} ^{p+q}(X, \mathbb {C} ) is q-opposite to its complex conjugate.
Since q=-p+(p+q), this proves the first part of (iii) of 507hodge-theory-ii-3.2.11hodge-theory-ii-3.2.11.xml3.2.11hodge-theory-ii-3.2
We now prove (ii) and (iii) of , which will finish the proof.
Independence from the choice of K^ \bullet.
The filtrations F and W of \operatorname {H} ^ \bullet (X, \mathbb {C} ) are the abutments of the hypercohomology spectral sequences of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle for the filtrations F and W.
These whole spectral sequences do not depend on the choice of K^ \bullet.
Functoriality.
Let f \colon X_1 \to X_2 be a morphism of schemes.
Suppose that we are given a morphism of smooth compactifications
505hodge-theory-ii-3.2.11.1hodge-theory-ii-3.2.11.1.xml3.2.11.1hodge-theory-ii-3.2.11 \begin {CD} X_1 @>f>> X_2 \\ @V{j_1}VV @VV{j_2}V \\ \bar {X}_1 @>>{ \bar {f}}> \bar {X}_2 \end {CD} \tag{3.2.11.1}
with the Y_i= \bar {X}_i \setminus X_i normal crossing divisors.
The canonical morphism (cf. ) from \bar {f}^* \Omega _{ \bar {X}_2}^ \bullet \langle Y_2 \rangle to \Omega _{ \bar {X}_1}^ \bullet \langle Y_1 \rangle is then a morphism of bifiltered complexes;
on hypercohomology, it induces a morphism compatible with F and W, and thus f^* \colon \operatorname {H} ^n(X_2, \mathbb {Z} ) \to \operatorname {H} ^n(X_1, \mathbb {Z} ) is a morphism of mixed Hodge structures, for the structures defined by the compactifications \bar {X}_i.
Independence from the choice of compactification.
With the notation of (B), if f is an isomorphism, then f^* is a bijective morphism of mixed Hodge structures, and thus an isomorphism ().
If \bar {X}_1 and \bar {X}_2 are two smooth compactifications of X, with Y_i= \bar {X}_i \setminus X smooth crossing divisors, then there exists a third smooth compactification \bar {X}, with Y= \bar {X} \setminus X a smooth crossing divisor, that fits into a commutative diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& X \ar [dl,hook] \ar [d,hook] \ar [dr,hook]
\\ \bar {X}_1
& \bar {X} \ar [l] \ar [r]
& \bar {X}_2
\end {tikzcd}
Namely, we can take \bar {X} to be a resolution of singularities of the closure of the diagonal image of X in \bar {X}_1 \times \bar {X}_2.
The identity map from \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_1 to \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_2 is thus the composite of two isomorphisms.
To finish the proof of (ii) and (iii), we note that every morphism f fits into a diagram of the form in : we choose compactifications \bar {X}'_1 and \bar {X}_2 of X_1 and X_2, then we take \bar {X}_1 to be a resolution of singularities of the closure of the image of X_1 in \bar {X}'_1 \times \bar {X}_2.
508Definitionhodge-theory-ii-3.2.12hodge-theory-ii-3.2.12.xml3.2.12hodge-theory-ii-3.2
The mixed Hodge structure of the cohomology of a smooth separated algebraic variety is the mixed Hodge structure from (iii) of .
515Corollaryhodge-theory-ii-3.2.13hodge-theory-ii-3.2.13.xml3.2.13hodge-theory-ii-3.2
With the above notation:
The spectral sequence in degenerates at E_2, i.e. the Leray spectral sequence for the inclusion j \colon X^* \hookrightarrow X degenerates at E_3 (i.e. E_3=E_ \infty).
The spectral sequence in
{}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} )
degenerates at E_1.
The spectral sequence defined by the sheaf \Omega _X^p \langle Y \rangle, endowed with the filtration W
\begin {aligned} E_1^{-n,k+n} &= \operatorname {H} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _X^p \langle Y \rangle )) \\ & \approx \operatorname {H} ^k( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^{p-n}( \varepsilon ^n)) \Rightarrow \operatorname {H} ^k( \bar {X}, \Omega _X^p \langle Y \rangle ) \end {aligned}
degenerates at E_2.
514Proofhodge-theory-ii-3.2.13
Claim (i) is proven in .
Consider the four spectral sequences in , with respect to the bifiltered complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
By (i), we have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum _{p,q} \dim {}_WE_2^{p,q}.
The {}_WE_1^{n,k-n} pages are, for fixed n, the abutments of a spectral sequence that degenerates at E_1 () with initial pages E_1^{p,n,k-p-n} (with the notation of ) the initial pages of the spectral sequence in (iii).
Since {}_Wd_1 is strictly compatible with the filtration F which is the abutment of this spectral sequence (), and is () the abutment of a morphism between these spectral sequences, which starts with the differentials from (iii), we have
\operatorname {Gr} _F^p({}_WE_2^{n,k-n}) \approx \operatorname {H} ^ \bullet (E_1^{p,n-1,k-n-p} \to E_1^{p,n,k-n-p} \to E_1^{p,n+1,k-n-p})
and \operatorname {Gr} _F^ \bullet ({}_WE_2^{ \bullet \bullet }) is the sum of the pages E_2^{ \bullet \bullet \bullet } of the spectral sequences in (iii).
We thus have
513Equationhodge-theory-ii-3.2.13.1hodge-theory-ii-3.2.13.1.xml3.2.13.1hodge-theory-ii-3.2.13 \sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum \dim E_2^{ \bullet \bullet \bullet }. \tag{3.2.13.1}
We also have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) \leqslant \dim {}_FE_1^{ \bullet \bullet }
with equality if and only if the spectral sequence in (ii) degenerates at E_1, and
\sum \dim {}_FE_1^{ \bullet \bullet } \leqslant \sum \dim E_2^{ \bullet \bullet \bullet }
with equality if and only if the spectral sequences in (iii) degenerate at E_2.
Comparing with , we obtain .
517Corollaryhodge-theory-ii-3.2.14hodge-theory-ii-3.2.14.xml3.2.14hodge-theory-ii-3.2
Let \omega be a meromorphic differential p-form on \bar {X} that is holomorphic on X and presents at worst logarithmic poles along Y.
Then the restriction \omega |X of \omega to X is closed, and if the cohomology class in \operatorname {H} ^p(X, \mathbb {C} ) defined by \omega is zero then \omega =0.
516Proofhodge-theory-ii-3.2.14
This is the particular case of (ii) of where {}_FE_1^{p0}={}_FE_ \infty ^{p0}.
522Corollaryhodge-theory-ii-3.2.15hodge-theory-ii-3.2.15.xml3.2.15hodge-theory-ii-3.2
If X is a smooth complete algebraic variety, then the mixed Hodge structure on \operatorname {H} ^n(X, \mathbb {Z} ) is the classical Hodge structure of weight n.
The Hodge numbers h^{pq} of the mixed Hodge structure of \operatorname {H} ^n(X, \mathbb {Z} ) (with X algebraic and smooth) can only be zero for p \leqslant n, q \leqslant n, and p+q \geqslant n
521Proofhodge-theory-ii-3.2.15
Claim (i) is clear;
to prove (ii), note that, if Y is the union of smooth divisors, then the rational Hodge structure \operatorname {Gr} _k^W( \operatorname {H} ^n(X, \mathbb {Q} )) is the quotient of a sub-object of a Hodge structure of weight n+k, namely
\operatorname {H} ^{n-k}( \widetilde {Y}^n, \mathbb {Q} ) \otimes \mathbb {Q} (-k).
523hodge-theory-ii-3.2.16hodge-theory-ii-3.2.16.xml3.2.16hodge-theory-ii-3.2
Let X be a smooth separated scheme.
We know that X admits smooth compactifications \bar {X} and that the subgroup of \operatorname {H} ^n(X, \mathbb {Z} ) given by the image of \operatorname {H} ^n( \bar {X}, \mathbb {Z} ) is independent of the choice of \bar {X} (cf. [G1968, (9.1) to (9.4)]).
525Corollaryhodge-theory-ii-3.2.17hodge-theory-ii-3.2.17.xml3.2.17hodge-theory-ii-3.2
Under the hypotheses of , the image of \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) in \operatorname {H} ^n(X, \mathbb {Q} ) is W_n( \operatorname {H} ^n(X, \mathbb {Q} )) (where W denotes the weight filtration ()).
524Proofhodge-theory-ii-3.2.17
We can suppose that \bar {X} \setminus X is a normal crossing divisor.
The claim then follows from the fact that W[-n] is the abutment of the Leray spectral sequence for the inclusion j \colon X \hookrightarrow \bar {X}.
527Corollaryhodge-theory-ii-3.2.18hodge-theory-ii-3.2.18.xml3.2.18hodge-theory-ii-3.2
Let f be a morphism from a smooth proper scheme Y to a smooth scheme X, with X admitting a smooth compactification j \colon X \hookrightarrow \bar {X}.
Then the groups \operatorname {H} ^n(X, \mathbb {Q} ) and \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) have the same image in \operatorname {H} ^n(Y, \mathbb {Q} ).
526Proofhodge-theory-ii-3.2.18
Since f^* and (jf)^* are strictly compatible with the weight filtration ((iii) of ), it suffices to prove that \operatorname {Gr} ^W(f^*) and \operatorname {Gr} ^W(f^*j^*) have the same image in \operatorname {Gr} ^W( \operatorname {H} ^n(Y, \mathbb {Q} )).
By and , \operatorname {Gr} _n^W(j^*) is an isomorphism, while \operatorname {Gr} _m^W(f^*)=0 for m \neq n since \operatorname {Gr} _m^W( \operatorname {H} ^n(Y, \mathbb {Q} ))=0 for m \neq n.
528Remarkhodge-theory-ii-3.2.19hodge-theory-ii-3.2.19.xml3.2.19hodge-theory-ii-3.2
By and , under the hypotheses of , the Hodge filtration on \operatorname {H} ^n(X, \mathbb {C} ) is the abutment of the hypercohomology spectral sequence of the complex j_*^m \Omega _{X^*}^ \bullet endowed with the filtration given by the pole-order filtration (): this spectral sequence coincides with .
1700hodge-theory-ii-4.4hodge-theory-ii-4.4.xmlHodge Theory II › Applications and supplements › Homomorphisms from abelian schemes4.4hodge-theory-ii-41701hodge-theory-ii-1.4hodge-theory-ii-1.4.xmlHodge Theory II › Filtrations › Hypercohomology of filtered complexes1.4hodge-theory-ii-1
In this section, we recall some standard constructions in hypercohomology.
We do not use the language of derived categories, which would be more natural here.
Throughout this entire section, by "complex" we mean "bounded-below complex".992hodge-theory-ii-1.4.1hodge-theory-ii-1.4.1.xml1.4.1hodge-theory-ii-1.4
Let T be a left-exact functor from an abelian category \mathscr {A} to an abelian category \mathscr {B}.
Suppose that every object of \mathscr {A} injects into an injective object;
the derived functors \mathrm {R} ^iT \colon \mathscr {A} \to \mathscr {B} are then defined.
An object A of \mathscr {A} is said to be *acyclic* for T if \mathrm {R} ^iT(A)=0 for i>0.
994hodge-theory-ii-1.4.2hodge-theory-ii-1.4.2.xml1.4.2hodge-theory-ii-1.4
Let (A,F) be a filtered object with finite filtration, and TF the filtration of TA by its sub-objects TF^p(A) (these are sub-objects since T is left exact).
If \operatorname {Gr} _F(A) is T-acyclic, then the F^p(A) are T-acyclic as successive extensions of T-acyclic objects.
The image under T of the sequence
0 \to F^{p+1}(A) \to F^p(A) \to \operatorname {Gr} ^p(A) \to 0
is thus exact, and
993Equationhodge-theory-ii-1.4.2.1hodge-theory-ii-1.4.2.1.xml1.4.2.1hodge-theory-ii-1.4.2 \operatorname {Gr} _{FT}TA \xrightarrow { \sim } T \operatorname {Gr} _FA. \tag{1.4.2.1} 995hodge-theory-ii-1.4.3hodge-theory-ii-1.4.3.xml1.4.3hodge-theory-ii-1.4
Let A be an object endowed with finite filtrations F and W such that \operatorname {Gr} _F \operatorname {Gr} _W A are T-acyclic.
The objects \operatorname {Gr} _FA and \operatorname {Gr} _WA are then T-acyclic, as well as the F^q(A) \cap W^p(A).
The sequences
0 \to T(F^q \cap W^{p+1}) \to T(F^q \cap W^p) \to T((F^q \cap W^p)/(F^q \cap W^{p+1})) \to 0
are thus exact, and T(F^q( \operatorname {Gr} _W^p(A))) is the image in T( \operatorname {Gr} _W^p(A)) of T(F^p \cap W^q).
The diagram
\begin {CD} T(F^q \cap W^p) @>>> T(F^q \operatorname {Gr} _W^pA) @>>> T \operatorname {Gr} _W^pA \\ @V{ \cong }VV @. @VV{ \cong }V \\ TF^q \cap TW^p @= TF^q \cap TW^p @>>> \operatorname {Gr} _{TW}^pTA \end {CD}
then shows that the isomorphism in relative to W sends the filtration \operatorname {Gr} _{TW}(TF) to the filtration T( \operatorname {Gr} _W(F)).
999hodge-theory-ii-1.4.4hodge-theory-ii-1.4.4.xml1.4.4hodge-theory-ii-1.4
Let K be a complex of objects of \mathscr {A}.
The hypercohomology objects \mathrm {R} ^iT(K) are calculated as follows:
We choose a quasi-isomorphism i \colon K \to K such that the components of K' are acyclic for T.
For example, we can take K' to be the simple complex associated to an injective Cartan–Eilenberg resolution of K.
We set
\mathrm {R} ^iT(K) = \operatorname {H} ^i(T(K')).
We can show that \mathrm {R} ^iT(K) does not depend on the choice of K', but depends functorially on K, and that a quasi-isomorphism f \colon K_1 \to K_2 induces *isomorphisms*
\mathrm {R} ^iT(f) \colon \mathrm {R} ^iT(K_1) \to \mathrm {R} ^iT(K_2). 1000hodge-theory-ii-1.4.5hodge-theory-ii-1.4.5.xml1.4.5hodge-theory-ii-1.4
Let F be a biregular filtration of K.
A {#T}-acyclic filtered resolution of K is a filtered quasi-isomorphism i \colon K \to K' from K to a filtered biregular complex such that the \operatorname {Gr} ^p({K'}^n) are acyclic for T.
If K' is such a resolution, then the {K'}^n are acyclic for T, and the filtered complex (cf. ) T(K') defines a spectral sequence
E_1^{pq} = \mathrm {R} ^{p+q}T( \operatorname {Gr} ^p(K)) \Rightarrow \mathrm {R} ^{p+q}T(K).
This is independent of the choice of K'.
We call this the hypercohomology spectral sequence of the filtered complex K.
It depends functorially on K, and a filtered quasi-isomorphism induces an isomorphism of spectral sequences.
The differentials d_1 of this spectral sequence are the connection morphisms defined by the short exact sequences
0 \to \operatorname {Gr} ^{p+1}K \to F^pK/F^{p+2}K \to \operatorname {Gr} ^pK \to 0. 1001hodge-theory-ii-1.4.6hodge-theory-ii-1.4.6.xml1.4.6hodge-theory-ii-1.4
Let K be a complex.
We denote by \tau _{ \leqslant p}(K) the following subcomplex:
\tau _{ \leqslant p}(K)^n = \begin {cases} K^n & \text {for }n<p \\ \operatorname {Ker} (d) & \text {for }n=p \\ 0 & \text {for }n>p. \end {cases}
The filtration, said to be canonical, of K by the \tau _{ \leqslant p}(K) is induced by shifting the trivial filtration G for which G^0(K)=K and G^1(K)=0.
We have, for the canonical filtration,
\begin {aligned} E_1^{pq} = 0 & \qquad \text {if }p+q \neq-p \\ \\ H^{-p} & \qquad \text {if }p+q=-p. \end {aligned}
A quasi-isomorphism f \colon K \to K' is automatically a filtered quasi-isomorphism for the canonical filtrations.
1002hodge-theory-ii-1.4.7hodge-theory-ii-1.4.7.xml1.4.7hodge-theory-ii-1.4
The subcomplexes \sigma _{ \geqslant p}(K) of K
\sigma _{ \geqslant p}(K)^n = \begin {cases} 0 & \text {if }n<p \\ K^n & \text {if }n \geqslant p \end {cases}
define a biregular filtration, called the stupid filtration of K.
The hypercohomology spectral sequences attached to the stupid or canonical filtrations of K are the two hypercohomology spectral sequences of K.
1003Examplehodge-theory-ii-1.4.8hodge-theory-ii-1.4.8.xml1.4.8hodge-theory-ii-1.4
Let f \colon X \to Y be a continuous map between topological spaces, and let \mathscr {F} be an abelian sheaf on X.
Let \mathscr {F}^ \bullet be a resolution of \mathscr {F} by f_*-acyclic sheaves.
We have \mathrm {R} ^i f_* \mathscr {F} \simeq \operatorname { \mathscr {H}} ^i(f_* \mathscr {F}^ \bullet ).
We take the functor T to be the functor \Gamma (Y,-).
The hypercohomology spectral sequence of the complex f_* \mathscr {F}^ \bullet endowed with its canonical filtration
E_1^{pq} = \operatorname {H} ^{2p+q}(Y, \mathrm {R} ^{-p}f_* \mathscr {F}) \Rightarrow \operatorname {H} ^{p+q}(X, \mathscr {F})
is exactly, up to the renumbering E_r^{pq} \mapsto E_{r+1}^{2p+q,-p}, the Leray spectral sequence for f and \mathscr {F}.
1010hodge-theory-ii-1.4.9hodge-theory-ii-1.4.9.xml1.4.9hodge-theory-ii-1.4
Let (K,W,F) be a biregular bifiltered complex.
To this complex, we associate:
A spectral sequence
{}_W E_1^{p,n-p} = \operatorname {H} ^n( \operatorname {Gr} _W^p(K)) \Rightarrow \operatorname {H} ^n(K)
with differentials {}_W d_1 being the connecting morphisms induced by the short exact sequences
0 \to \operatorname {Gr} _W^{p+1}(K) \to W^p(K)/W^{p+2}(K) \to \operatorname {Gr} _W^p(K) \to 0;
An analogous spectral sequence for the filtration F;
Exact squares
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& 0 \dar
& 0 \dar
& 0 \dar
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^{q+1}K \rar \dar
& F^p/F^{p+2}( \operatorname {Gr} _W^{q+1}K) \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1}K \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1}(W^q/W^{q+2}(K)) \rar \dar
& F^p/F^{p+2}(W^q/W^{q+2}(K)) \rar \dar
& \operatorname {Gr} _F^p(W^q/W^{q+2}(K)) \rar \dar
& 0
\\ 0 \rar
& \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \rar \dar
& F^p/F^{p+2} \operatorname {Gr} _W^q K \rar \dar
& \operatorname {Gr} _F^p \operatorname {Gr} _W^q K \rar \dar
& 0
\\ & 0
& 0
& 0
\end {tikzcd}
The exterior rows and columns of this square define connection morphisms
\begin {aligned} {}_{F,W}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^{p+1} \operatorname {Gr} _W^q K \\ {}_{W,F}d_1 \colon \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K & \to \operatorname {H} ^{n+1} \operatorname {Gr} _F^p \operatorname {Gr} _W^{q+1} K. \end {aligned}
These morphisms satisfy
\begin {gathered} {}_{FW}d_1 \circ {}_{WF}d_1 + {}_{WF}d_1 \circ {}_{FW}d_1 = 0 \\ {}_{FW}d_1^2 = 0 \\ {}_{WF}d_1^2 = 0 \end {gathered}
The morphisms {}_{FW}d_1 are the morphisms d_1 of the spectral sequences E^{(q)}, with the E_1^{(q)p,n-p} term equal to
1008Equationhodge-theory-ii-1.4.9.1hodge-theory-ii-1.4.9.1.xml1.4.9.1hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _W^q K) = {}_WE_1^{q,n-q} \tag{1.4.9.1}
defined by the filtered complex \operatorname {Gr} _W^q(K).
This spectral sequence abuts to the filtration induced by F on \operatorname {H} ^ \bullet \operatorname {Gr} _W^q K.
Similarly, the {}_{WF}d_1 are the d_1 of spectral sequences with the same initial terms
1009Equationhodge-theory-ii-1.4.9.2hodge-theory-ii-1.4.9.2.xml1.4.9.2hodge-theory-ii-1.4.9 E_1^{p,q,n-p-q} \coloneqq \operatorname {H} ^n( \operatorname {Gr} _F^p \operatorname {Gr} _W^q K) \Rightarrow \operatorname {H} ^n( \operatorname {Gr} _F^p K). \tag{1.4.9.1}
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& \operatorname {H} ^n \operatorname {Gr} _W^q K
\ar [Rightarrow,dr,"{}_Wd_1"]
\\ \operatorname {H} ^n \operatorname {Gr} _F^p \operatorname {Gr} _W^q K
\ar [Rightarrow,ur,"{}_{F,W}d_1"]
\ar [Rightarrow,dr,swap,"{}_{W,F}d_1"]
&& \operatorname {H} ^n K
\\ & \operatorname {H} ^n \operatorname {Gr} _F^p K
\ar [Rightarrow,ur,swap,"{}_Fd_1"]
\end {tikzcd}
These constructions are symmetric in F and W via the isomorphism
\operatorname {Gr} _F^p \operatorname {Gr} _W^q \sim \operatorname {Gr} _W^q \operatorname {Gr} _F^p. 1011hodge-theory-ii-1.4.10hodge-theory-ii-1.4.10.xml1.4.10hodge-theory-ii-1.4
We can also interpret the {}_{W,F}d_1 are the initial morphisms of a morphism of spectral sequences , abutting to {}_Wd_1.
Indeed, let C^q be the cone of the morphism
W^q(K)/W^{q+2}(K) \to \operatorname {Gr} _W^q(K).
In the diagram
\Sigma \colon \operatorname {Gr} _W^{q+1}(K)[1] \xrightarrow {u} C^q \xleftarrow {i} \operatorname {Gr} _W^q(K)
the morphism u is a quasi-isomorphism, and we have
{}_Wd_1 = \operatorname {H} (u)^{-1} \circ \operatorname {H} (i).
In fact, u is even a filtered (for F) quasi-isomorphism, and the above construction defines a morphism from the spectral sequence defined by ( \operatorname {Gr} _W^q(K),F) to that defined by ( \operatorname {Gr} _W^{q+1}(K)[1],F), and it abuts to {}_Wd_1.
The initial term of this morphism, induced by \operatorname {Gr} _F( \Sigma ) is exactly {}_{W,F}d_1.
1012hodge-theory-ii-1.4.11hodge-theory-ii-1.4.11.xml1.4.11hodge-theory-ii-1.4
These constructions pass as they are to hypercohomology.
Let K be a complex endowed with two biregular filtrations F and W.
A bifiltered T-acyclic resolution of K is a bifiltered quasi-isomorphism i \colon K \to K' such that the \operatorname {Gr} _F^p \operatorname {Gr} _W^n({K'}^m) are T-acyclic.
Such a morphism always exists.
In the particular case where \mathscr {A} is the category of sheaves of A-modules on a topological space X, and where T is the functor \Gamma from \mathscr {A} to the category of A-modules, then an example of a bifiltered T-acyclic resolution of K is the simple complex associated to the double complex given by the Godement resolution \mathscr {C}^ \bullet (K) of K, filtered by the \mathscr {C}^ \bullet (F^p(K)) and the \mathscr {C}^ \bullet (W^n(K)).
Since \mathscr {C}^ \bullet is exact, we have
\operatorname {Gr} _F \operatorname {Gr} _W( \mathscr {C}^ \bullet (K)) \simeq \mathscr {C}^ \bullet ( \operatorname {Gr} _F \operatorname {Gr} _W(K)).
We will not have need here of any other case.
If K' is a bifiltered T-acyclic resolution of K, then the complex TK' is filtered by the TF^pK' and the TW^qK' ().
Furthermore, \operatorname {Gr} _W^n(K') is a T-acyclic filtered (for F) resolution of \operatorname {Gr} _F^n(K), and \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K') is a T-acyclic resolution of \operatorname {Gr} _F^n \operatorname {Gr} _W^m(K), and
\begin {gathered} T \operatorname {Gr} _F K' \approx \operatorname {Gr} _F TK' \qquad \text {as }W \text {-filtered complexes} \\ T \operatorname {Gr} _W K' \approx \operatorname {Gr} _W TK' \qquad \text {as }F \text {-filtered complexes} \\ T \operatorname {Gr} _F \operatorname {Gr} _W K' \approx \operatorname {Gr} _F \operatorname {Gr} _W TK'. \end {gathered} 1013Lemmahodge-theory-ii-1.4.12hodge-theory-ii-1.4.12.xml1.4.12hodge-theory-ii-1.4
Under the hypotheses of :
The initial terms of the hypercohomology spectral sequences
{}_WE_1^{q,n-q} = \mathrm {R} ^nT( \operatorname {Gr} _W^qK) \Rightarrow \mathrm {R} ^nT(K) \tag{1}
{}_FE_1^{p,n-p} = \mathrm {R} ^nT( \operatorname {Gr} _F^pK) \Rightarrow \mathrm {R} ^nT(K) \tag{2}
are abutments of the hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _W^qK and \operatorname {Gr} _F^pK, with E_1 pages given by
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_WE_1^{q,n-q} \qquad \text {for fixed }q \tag{3}
E_1^{p,q,n-p-q} \coloneqq \mathrm {R} ^nT( \operatorname {Gr} _F^p \operatorname {Gr} _W^qK) \Rightarrow {}_FE_1^{p,n-p} \qquad \text {for fixed }p \tag{4}
The filtration of {}_WE_1^{p,n-p}, abutting to the spectral sequence (3), is the filtration of {}_WE_1^{p,n-p}(TK') induced by the filtration F of TK'.
For the differentials of the complexes \operatorname {Gr} _W^n(T(K')) to be strictly compatible with the filtration F, it is necessary and sufficient that the hypercohomology spectral sequences (3) degenerate on the E_1 page.
The morphisms d_1 of the spectral sequence (4) are the initial terms of the degree-1 morphisms of the spectral sequences (3) that abut to the morphisms d_1 of the spectral sequence (1).
895Proofhodge-theory-ii-1.4.12
Claims (i) and (iv) follow from and applied to TK' via the isomorphisms .
Claim (ii) is then trivial by the definition of the recurrent filtration F (identical to the discrete filtrations by and (iii) of ), and claim (iii) follows from .
1702hodge-theory-ii-introductionhodge-theory-ii-introduction.xmlHodge Theory II › Introductionhodge-theory-iiWork presented as a doctoral thesis at l'Université d'Orsay.
By Hodge, the cohomology space \operatorname {H} ^n(X, \mathbb {C} ) of a compact Kähler variety X is endowed with a "Hodge structure" of weight n, i.e. a natural bigrading
\operatorname {H} ^n(X, \mathbb {C} ) = \bigoplus _{p+q=n} \operatorname {H} ^{p,q}
which satisfies \overline { \operatorname {H} ^{p,q}}= \operatorname {H} ^{q,p}.
We will show here that the complex cohomology of a non-singular, not necessarily compact, algebraic variety is endowed with a structure of a slightly more general type, which presents \operatorname {H} ^n(X, \mathbb {C} ) as a "successive extension" of Hodge structures of decreasing weights, contained between 2n and n, whose Hodge numbers h^{p,q}= \dim \operatorname {H} ^{p,q}, are zero for both p>n and q>n.
The reader will find an explanation in [Hodge Theory I] of the yoga that underlies this construction.
The proof, which is essentially algebraic, relies on one hand on Hodge theory, and on the other on Hironaka's resolution of singularities, which allows us, via a spectral sequence, to "express" the cohomology of a non-singular quasi-projective algebraic variety in terms of the cohomology of non-singular projective varieties.
contains, apart from reminders on filtrations gathered together for the ease of the reader, two key results:
, which will only be used via its corollary, [hodge-theory-ii-2.3.5 (?)], which gives the fundamental properties of "mixed Hodge structures".
The "two filtrations lemma", [hodge-theory-ii-1.3.16 (?)].
[hodge-theory-ii-2 (?)] recalls Hodge theory and introduces mixed Hodge structures.
The heart of this work is [hodge-theory-ii-3.2 (?)], which defines the mixed Hodge structure of \operatorname {H} ^n(X, \mathbb {C} ), and establishes some degenerations of spectral sequences.
[hodge-theory-ii-4 (?)] gives diverse applications, all following from [hodge-theory-ii-4.1.1 (?)] and the theory of the (K/k)-trace, for the resulting Hodge structures ([hodge-theory-ii-4.1.2 (?)]).
The principal ones are [hodge-theory-ii-4.2.6 (?)] and [hodge-theory-ii-4.4.15 (?)].
1703hodge-theory-ii-3.1hodge-theory-ii-3.1.xmlHodge Theory II › Hodge theory of non-singular algebraic varieties › Logarithmic poles and residues3.1hodge-theory-ii-31254hodge-theory-ii-3.1.1hodge-theory-ii-3.1.1.xml3.1.1hodge-theory-ii-3.1
We recall several classical properties of "logarithmic poles" of holomorphic differential forms.
The reader will find proofs in [D1970, II, (3.1) to (3.7)], for example.
1255hodge-theory-ii-3.1.2hodge-theory-ii-3.1.2.xml3.1.2hodge-theory-ii-3.1
A divisor Y in a smooth complex analytic variety X is said to be of normal crossing if the inclusion of Y into X is locally isomorphic to the inclusion of a union of coordinate hyperplanes in \mathbb {C} ^n;
this does not imply that Y is the union of smooth divisors.
Let Y be a normal crossing divisor in X, and j the inclusion of X^*=X \setminus Y into X.
We denote by \Omega _X^1 \langle Y \rangle the locally free sub-\mathcal {O}-module of j_* \Omega _{X^*}^1 generated by \Omega _X^1 and the \mathrm {d} z_i/z_i for z_i local coordinates of a local irreducible component of Y.
The sheaf \Omega _X^p \langle Y \rangle of differential p-forms on X with logarithmic poles along Y is by the definition the locally free sub-sheaf \wedge ^p \Omega _X^1 \langle Y \rangle of j_* \Omega _{X^*}^p.
1259Propositionhodge-theory-ii-3.1.3hodge-theory-ii-3.1.3.xml3.1.3hodge-theory-ii-3.1
A section \alpha of j_* \Omega _{X^*}^p belongs to \Omega _X^p \langle Y \rangle if and only if \alpha and \mathrm {d} \alpha have at worst simple poles along the divisor Y.
The \Omega _X^p \langle Y \rangle form the smallest subcomplex of j_* \Omega _{X^*}^ \bullet that is stable under exterior product and that contains \Omega _X^ \bullet and the logarithmic differential \mathrm {d} f/f of every local section of j_* \Omega _{X^*}^p meromorphic along Y.
We call \Omega _X^ \bullet \langle Y \rangle the logarithmic de Rham complex of X along Y.
By (ii) of , this complex is contravariant in the pair (X,X^*).
1261hodge-theory-ii-3.1.4hodge-theory-ii-3.1.4.xml3.1.4hodge-theory-ii-3.1
Locally on X, Y is the union of various smooth divisors Y_i, and we denote by Y^n (resp. \widetilde {Y}^n) the union (resp. disjoint sum) of the n-fold intersections of the Y_i.
The Y^n glue to give a subspace Y^n of X, and the \widetilde {Y}^n glue to give the normalisation variety of Y^n.
We have \widetilde {Y}^0=Y^0=X, and we set \widetilde {Y}= \widetilde {Y}^1.
We define the two-element set of orientations of an n-element set E as the set of generators of \wedge ^n \mathbb {Z} ^E.
For n \geqslant2, this set is that of the conjugation classes under the alternating group of total orders on E.
If, to each point y of \widetilde {Y}^n, we associate the set of n local components of Y that contain the image in X of a neighbourhood of y in \widetilde {Y}^n, then we define on \widetilde {Y}^n a local system E_n of n-element sets.
The local system of orientations of these sets is a \mathbb {Z} {/}(2)-torsor.
This torsor defines, via the inclusion of \mathbb {Z} {/}(2) into \mathbb {C} ^*, a complex local system \varepsilon ^n of rank 1 on \widetilde {Y}^n, endowed with an isomorphism ( \varepsilon ^n)^{ \otimes2 } \simeq \mathbb {C}.
We have
\varepsilon ^n \simeq \bigwedge ^n \mathbb {C} ^{E_n}.
Locally on \widetilde {Y}^n, \varepsilon ^n is endowed with two opposite isomorphisms \pm \alpha \colon \varepsilon ^n \xrightarrow { \sim } \mathbb {C}.
We set
\varepsilon _ \mathbb {Z} ^n = \alpha ^{-1}((2 \pi i)^{-n} \mathbb {Z} ).
There is a way of seeing \varepsilon ^n, endowed with \varepsilon _ \mathbb {Z} ^n, as a twisted version of \mathbb {Z} (-n) over \widetilde {Y}^n.
We denote by \varepsilon _X^n (resp. by ( \varepsilon _X^n)_ \mathbb {Z}) the direct image of \varepsilon ^n (resp. of \varepsilon _ \mathbb {Z} ^n) under the mat from \widetilde {Y}^n to X.
We have
1260Equationhodge-theory-ii-3.1.4.1hodge-theory-ii-3.1.4.1.xml3.1.4.1hodge-theory-ii-3.1.4 \varepsilon _X^n \simeq \bigwedge ^n \varepsilon _X^1 \qquad \text {for }n \geqslant0 . \tag{3.1.4.1}
If Y is the union of distinct smooth divisors (Y_i)_{i \in I}, then the choice of a total order on I trivialises the \varepsilon ^n.
1263hodge-theory-ii-3.1.5hodge-theory-ii-3.1.5.xml3.1.5hodge-theory-ii-3.1
We denote by W_n( \Omega _X^p \langle Y \rangle ) the sub-module of \Omega _X^p \langle Y \rangle consisting of linear combinations of products
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \qquad \text {for }m \leqslant n
with \alpha holomorphic and the t_{i(j)} local coordinates of the distinct local components Y_j of Y.
We define the weight filtration of \Omega _X^ \bullet \langle Y \rangle to be the increasing filtration by the subcomplexes W_n( \Omega _X^ \bullet \langle Y \rangle ).
We have
1262Equationhodge-theory-ii-3.1.5.1hodge-theory-ii-3.1.5.1.xml3.1.5.1hodge-theory-ii-3.1.5 W_n( \Omega _X^p \langle Y \rangle ) \wedge W_m( \Omega _X^p \langle Y \rangle ) \subset W_{n+m}( \Omega _X^{p+q} \langle Y \rangle ). \tag{3.1.5.1}
If we denote by i_n the map from \widetilde {Y}^n to X, then we can show that the mapping
\alpha \wedge \frac { \mathrm {d} t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac { \mathrm {d} t_{i(m)}}{t_{i(m)}} \longmapsto ( \alpha |Y_{i(1)} \cap \ldots \cap Y_{i(n)}) \otimes ( \text {orientation }i(1) \ldots i(n))
defines an isomorphism of complexes
\operatorname {Res} \colon \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle ) \approx (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n]
(the Poincaré residue).
1264hodge-theory-ii-3.1.6hodge-theory-ii-3.1.6.xml3.1.6hodge-theory-ii-3.1
The interpretation that follows, in terms of [hodge-theory-ii-3.1.5.2 (?)], of the Leray spectral sequence for the inclusion of X^* into X, was pointed out to me by N. Katz.
It will allow us to prove a point that I had initially considered as evident (the first part of (ii) of [hodge-theory-ii-3.2.5 (?)]).
1265hodge-theory-ii-3.1.7hodge-theory-ii-3.1.7.xml3.1.7hodge-theory-ii-3.1
Every point of X admits a fundamental system of Stein open neighbourhoods whose intersections on X^* are again Stein.
For a coherent analytic sheaf \mathscr { F } on X^*, we thus have \mathrm {R} ^i j_* \mathscr { F } =0 for i>0.
The de Rham complex \Omega _{X^*}^ \bullet is thus a resolution of the constant sheaf \mathbb {C} by sheaves acyclic for the functor j_*.
Then
1056Equationhodge-theory-ii-3.1.7.1hodge-theory-ii-3.1.7.1.xml3.1.7.1hodge-theory-ii-3.1.7 \operatorname {H} ^ \bullet (X^*, \mathbb {C} ) \xrightarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X^*, \Omega _{X^*}^ \bullet ) \xleftarrow { \sim } \operatorname { \mathbb {H}} ^ \bullet (X,j_* \Omega _{X^*}^ \bullet )
and the Leray spectral sequence for the morphism j can be identified with the hypercohomology spectral sequence for j_* \Omega _{X^*}^ \bullet corresponding to the filtration \tau by the subcomplexes \tau _{ \leqslant -n}(j_* \Omega _{X^*}^ \bullet ) ().
1266Propositionhodge-theory-ii-3.1.8hodge-theory-ii-3.1.8.xml3.1.8hodge-theory-ii-3.1
The morphisms of filtered complexes
( \Omega _X^ \bullet \langle Y \rangle , W) \xleftarrow { \alpha } ( \Omega _X^ \bullet \langle Y \rangle , \tau ) \xhookrightarrow { \beta } (j_* \Omega _{X^*}^ \bullet , \tau )
are filtered quasi-isomorphisms.
They define an isomorphism between the Leray spectral sequence for j in complex cohomology and the hypercohomology spectral sequence of the filtered complex ( \Omega _X^ \bullet \langle Y \rangle , W) on X.
257Proofhodge-theory-ii-3.1.8
By and , it suffices to prove the first claim.
In [D1970, II, (6.9)] or [AH1955], one can find a proof of the fact that \beta is a quasi-isomorphism, and thus a filtered quasi-isomorphism.
We can also directly calculate the cohomology sheaves of the two sides: those of \Omega _X^ \bullet \langle Y \rangle are determined by [hodge-theory-ii-3.1.5.2 (?)], while those of j_* \Omega _{X^*}^ \bullet are the \mathrm {R} ^ij_* \mathbb {C}, which can be calculated by topological methods.
For n \geqslant p, we have
W_n( \Omega _X^p \langle Y \rangle ) = \Omega _X^p \langle Y \rangle
and so \alpha is a morphism from \Omega _X^ \bullet \langle Y \rangle, endowed with \tau, to \Omega _X^ \bullet \langle Y \rangle, endowed with the decreasing filtration associated to W ().
By [hodge-theory-ii-3.1.5.2 (?)], we have
256Equationhodge-theory-ii-3.1.8.1hodge-theory-ii-3.1.8.1.xml3.1.8.1hodge-theory-ii-3.1.8 \operatorname { \mathscr {H}} ^i( \operatorname {Gr} _n^W( \Omega _X^ \bullet \langle Y \rangle )) = \begin {cases} 0 & \text {for }i \neq n \\ \varepsilon _X^n & \text {for }i=n \end {cases} \tag{3.1.8.1}
and we deduce from the first line of this equation that \alpha is a filtered quasi-isomorphism.
This proves .
By , the isomorphism defines an isomorphism
258Equationhodge-theory-ii-3.1.8.2hodge-theory-ii-3.1.8.2.xml3.1.8.2hodge-theory-ii-3.1.8 \mathrm {R} ^n j_* \mathbb {C} \simeq \operatorname { \mathscr {H}} ^n(j_* \Omega _{X^*}^ \bullet ) \simeq \operatorname { \mathscr {H}} ^n( \Omega _X^ \bullet \langle Y \rangle ) \simeq \varepsilon _X^n. \tag{3.1.8.2}
The isomorphisms correspond, via , to the cup product.
1268Propositionhodge-theory-ii-3.1.9hodge-theory-ii-3.1.9.xml3.1.9hodge-theory-ii-3.1
The canonical morphism from \mathrm {R} ^nj_* \mathbb {Z} to \mathrm {R} ^nj_* \mathbb {C} identifies, via , the sheaf \mathrm {R} ^nj_* \mathbb {Z} with ( \varepsilon _X^n)_ \mathbb {Z} ().
1267Proofhodge-theory-ii-3.1.9
The question is local on X.
We can thus suppose that X is an open polycylinder D^m, with
D = \{ z \in \mathbb {C} \mid |z|<1 \}
and also that Y= \bigcup _{k=1}^ \ell Y_k with Y_k= \operatorname {pr} _k^{-1}(0).
The fibre at 0 of \mathrm {R} ^nj_* \mathbb {Z} is then the integer cohomology of X^*=(D^*)^ \ell \times D^{m- \ell }, with
D^* = \{ z \in \mathbb {C} \mid 0<|z|<1 \} .
The space X^* has the homotopy type of a torus;
its cohomology is thus torsion free, and the cup product defines isomorphisms
\bigwedge ^n( \mathrm {R} ^1j_* \mathbb {Z} ) \xrightarrow { \sim } ( \mathrm {R} ^nj_* \mathbb {Z} )_0.
It thus suffices to prove for n=1.
The integer homology \operatorname {H} _1(X^*) is generated by the loops \gamma _k that go around the various Y_k.
We have
\oint _{ \gamma _k} \frac { \mathrm {d} z_k}{z_k} = \pm2 \pi i
and so the integer cohomology is generated by the \frac {1}{2 \pi i} \frac { \mathrm {d} z_k}{z_k}, and this proves .
1271hodge-theory-ii-3.1.10hodge-theory-ii-3.1.10.xml3.1.10hodge-theory-ii-3.1
Let \mathscr { F } be a coherent analytic sheaf on X^*, given as the restriction to X^* of a coherent analytic sheaf \mathscr { F } ' on X.
We define the meromorphic direct image j_*^ \mathrm {m} \mathscr { F } of \mathscr { F } to be the inductive limit
j_*^ \mathrm {m} \mathscr { F } = \varinjlim \mathscr { F } '(nY).
Locally on X, Y is the sum of a finite family (Y_i)_{i \in I} of smooth divisors, and we define the pole-order filtration P on j_*^ \mathrm {m} \mathcal {O} _X^* by the equation
1269Equationhodge-theory-ii-3.1.10.1hodge-theory-ii-3.1.10.1.xml3.1.10.1hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \mathcal {O} _{X^*}) = \sum _{n \in A_p} \mathcal {O} _X \left ( \sum (n_i+1)Y_i \right ) \tag{3.1.10.1}
where
A_p = \left \{ (n_i)_{i \in I} \mid \sum _i n_i \leqslant -p \text { and } n_i \geqslant0 \text { for all } i \right \} .
This construction globalises by endowing j_*^ \mathrm {m} \mathcal {O} _{X^*} with an exhaustive filtration such that P^p=0 for p>0.
We define the pole-order filtration of the complex j_*^ \mathrm {m} \Omega _{X^*}^ \bullet =j_*^ \mathrm {m} \mathcal {O} _{X^*} \otimes \Omega _X^ \bullet to be the filtration
1270Equationhodge-theory-ii-3.1.10.2hodge-theory-ii-3.1.10.2.xml3.1.10.2hodge-theory-ii-3.1.10 P^p(j_*^ \mathrm {m} \Omega _{X^*}^k) = P^{p-k}(j_*^ \mathrm {m} \mathcal {O} _X) \otimes \Omega _X^k. \tag{3.1.10.2}
The filtration P induces, on the subcomplex \Omega _X^ \bullet \langle Y \rangle of j_*^ \mathrm {m} \Omega _{X^*}^ \bullet, the stupid filtration by the \sigma _{ \geqslant p}( \Omega _X^ \bullet \langle Y \rangle ), which we also call the Hodge filtration F.
1272Propositionhodge-theory-ii-3.1.11hodge-theory-ii-3.1.11.xml3.1.11hodge-theory-ii-3.1
The inclusion morphism
( \Omega _X^ \bullet \langle Y \rangle , F) \to (j_*^ \mathrm {m} \Omega _{X_*}^ \bullet , P)
is a filtered quasi-isomorphism.
254Proofhodge-theory-ii-3.1.11
This statement was suggested to me by [G1969].
A proof can be found in [D1970, II, (3.13)].
1704hodge-theory-ii-3.2hodge-theory-ii-3.2.xmlHodge Theory II › Hodge theory of non-singular algebraic varieties › Mixed Hodge theory3.2hodge-theory-ii-3
We recall that, from here on, we say "scheme" to mean a scheme of finite type over \mathbb {C}, and "sheaf on S" to mean a sheaf on S^ \mathrm {an}.
484hodge-theory-ii-3.2.1hodge-theory-ii-3.2.1.xml3.2.1hodge-theory-ii-3.2
Let X be a smooth and separated scheme.
By Nagata [N1962], X is a Zariski open of a complete scheme \bar {X}.
By Hironaka [H1964], we can take \bar {X} to be smooth and such that Y= \bar {X} \setminus X is a normal crossing divisor.
The reader who wishes to avoid the reference to Nagata can suppose X to be quasi-projective.
The smooth completion \bar {X} can then be chosen to be projective and such that Y is the union of smooth divisors.
If we limit ourselves to such compactifications, then we only need Hodge theory in its standard form ().
485hodge-theory-ii-3.2.2hodge-theory-ii-3.2.2.xml3.2.2hodge-theory-ii-3.2
By and , we have
\operatorname {H} ^ \bullet (X, \mathbb {C} ) \simeq \operatorname { \mathbb {H}} ^ \bullet ( \bar {X}, \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ).
We define the Hodge filtration F on the complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle as the filtration F^p= \sigma _{ \geqslant p} given by the stupid truncations ().
On \Omega _{ \bar {X}}^ \bullet \langle Y \rangle, we thus have two filtrations: F and W ().
489hodge-theory-ii-3.2.3hodge-theory-ii-3.2.3.xml3.2.3hodge-theory-ii-3.2
We will need to make use of the fact that there exist bifiltered resolutions i \colon \Omega _{ \bar {X}}^ \bullet \langle Y \rangle \to K^ \bullet such that the \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j) are \Gamma-acyclic sheaves:
\operatorname {H} ^i( \bar {X}, \operatorname {Gr} _F^p \operatorname {Gr} _n^W(K^j)) = 0 \qquad \text {for }i>0.
Here are two methods to construct such a resolution:
We can take K^ \bullet to be the canonical Godement resolution \mathscr {C}^ \bullet ( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), filtered by the \mathscr {C}^ \bullet (W_n( \Omega _{ \bar {X}} \langle Y \rangle )) and the \mathscr {C}^ \bullet (F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )).
This is a bifiltered resolution since \mathscr {C}^ \bullet is an exact functor.
We can take K^ \bullet to be the \mathrm {d} ''-resolution of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
Let \Omega _{ \bar {X}}^{pq} be the sheaf of C^ \infty forms of type (p,q);
then K^ \bullet is the simple complex associated to the double complex of the \Omega _{ \bar {X}}^p \langle Y \rangle \otimes _ \mathcal {O} \Omega _{ \bar {X}}^{0,q} (a subcomplex of the j_* \Omega _X^{ \bullet \bullet }).
This complex is filtered by the F^p( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet } and by the W_n( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \otimes \Omega _{ \bar {X}}^{0, \bullet };
to prove that this is a bifiltered resolution, we use the fact that the sheaf \mathcal {O} _ \infty of complex C^ \infty functions on \bar {X} is flat over \mathcal {O} (a corollary of the Malgrange C^ \infty preparation theorem).
The sheaves \operatorname {Gr} _F \operatorname {Gr} _W(K^ \bullet ) are fine, since they are sheaves of modules over the soft sheaf \mathcal {O} _ \infty.
492hodge-theory-ii-3.2.4hodge-theory-ii-3.2.4.xml3.2.4hodge-theory-ii-3.2
With the notation of , the complex cohomology of X appears as the cohomology of the bifiltered complex \Gamma ( \bar {X},K^ \bullet ).
We thus have two spectral sequences abutting to \operatorname {H} ^ \bullet (X, \mathbb {C} ).
They can be written, with the notation of , as:
490Equationhodge-theory-ii-3.2.4.1hodge-theory-ii-3.2.4.1.xml3.2.4.1hodge-theory-ii-3.2.4 {}_WE_1^{pq} = \operatorname { \mathbb {H}} ^{p+q}( \bar {X}, \varepsilon _{ \bar {X}}^{-p}[p]) = \operatorname { \mathbb {H}} ^{2p+q}( \widetilde {Y}^p, \varepsilon ^{-q}) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ) \tag{3.2.4.1}
491Equationhodge-theory-ii-3.2.4.2hodge-theory-ii-3.2.4.2.xml3.2.4.2hodge-theory-ii-3.2.4 {}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^n(X, \mathbb {C} ). \tag{3.2.4.2}
The first of these, up to the renumbering {}_WE_1^{pq} \mapsto E_2^{2p+q,-p}, is exactly the Leray spectral sequence of the inclusion j.
493Theoremhodge-theory-ii-3.2.5hodge-theory-ii-3.2.5.xml3.2.5hodge-theory-ii-3.2
On the pages {}_WE_r^{pq} of the spectral sequence , the first direct filtration, the second direct filtration, and the recurrent filtration defined by F all coincide.
The filtration on \operatorname {H} ^n(X, \mathbb {C} ) that is the abutment of the spectral sequence {}_WE comes from a filtration W of \operatorname {H} ^n(X, \mathbb {Q} ).
Neither it, nor the filtration F that is the abutment of the spectral sequence {}_FE, depend on the choice of compactification \bar {X} of X or on the choice of K^ \bullet.
The filtrations W[n] () and F define on \operatorname {H} ^n(X, \mathbb {Z} ) a mixed Hodge structure, functorially in X.
By , the spectral sequence {}_WE is the Leray spectral sequence for j_* (up to renumbering).
It is thus induced by tensoring a \mathbb {Q}-vectorial spectral sequence with \mathbb {C}, and so the first claim of (ii) is true.
Complex conjugation acts on the {}_WE;
it can be calculated via .
494Lemmahodge-theory-ii-3.2.6hodge-theory-ii-3.2.6.xml3.2.6hodge-theory-ii-3.2
The hypercohomology spectral sequences of the filtered complexes \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ), endowed with the filtration induced by the Hodge filtration, degenerate at the E_1 page.
369Proofhodge-theory-ii-3.2.6
Define the Y^n and the \widetilde {Y}^n as in , and let i_n \colon \widetilde {Y}^n \to X.
By [hodge-theory-ii-3.1.5.2 (?)], we have
\operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ) \sim (i_n)_* \Omega _{ \widetilde {Y}^n}^ \bullet ( \varepsilon ^n)[-n].
Furthermore, the Hodge filtration induces the stupid filtration (), and so the spectral sequence in is given by translations of the degrees of the classical spectral sequence
E_1^{pq} = \operatorname {H} ^q( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^p( \varepsilon ^n)) \Rightarrow \operatorname {H} ^{p+q}( \widetilde {Y}^n, \varepsilon ^n).
If \bar {X} is projective and Y the union of smooth divisors, then \widetilde {Y}^n is projective, \varepsilon ^n is a trivial local system, and classical Hodge theory () gives the degeneration claimed in .
For the general case, refer to [D1968] (see also and ).
Hodge theory also tells us that the Hodge filtration on \operatorname {H} ^k( \widetilde {Y}^n, \varepsilon ^n) is k-opposite to its complex conjugate (the complex conjugate being defined in terms of \varepsilon _ \mathbb {Z} ^n ()).
We have, taking into account the translations of the degrees:
495Lemmahodge-theory-ii-3.2.7hodge-theory-ii-3.2.7.xml3.2.7hodge-theory-ii-3.2
The filtration on
{}_WE_1^{-n,k+n} = \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \approx \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n)
which is the abutment of the spectral sequence in is (k+n)-opposite to its complex conjugate.
498Lemmahodge-theory-ii-3.2.8hodge-theory-ii-3.2.8.xml3.2.8hodge-theory-ii-3.2
The differentials d_1 of the spectral sequence {}_WE are strictly compatible with the filtration F.
497Proofhodge-theory-ii-3.2.8
On the E_1 pages, there is only the filtration induced by F to consider ( and (iii) of ), and d_1 is compatible with this filtration ((i) of ).
This filtration is the abutment of the spectral sequence in ((ii) of ).
By , the arrow
d_1 \colon \operatorname { \mathbb {H}} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle )) \to \operatorname { \mathbb {H}} ^{k+1}( \bar {X}, \operatorname {Gr} _{n-1}^W( \Omega _{ \bar {X}}^ \bullet \langle Y \rangle ))
or
496Equationhodge-theory-ii-3.2.8.1hodge-theory-ii-3.2.8.1.xml3.2.8.1hodge-theory-ii-3.2.8 d_1 \colon \operatorname {H} ^{k-n}( \widetilde {Y}^n, \varepsilon ^n) \to \operatorname {H} ^{k-n+1}( \widetilde {Y}^{n-1}, \varepsilon ^{n-1}) \tag{3.2.8.1}
is compatible with the filtrations that are (k+n)-opposite to their complex conjugate.
Since d_1 commutes up to complex conjugation, d_1 respects the bigrading (of weight k+n) defined by F and \bar {F}, which proves .
Furthermore, the cohomology of the complex E_1 is again bigraded:
499Lemmahodge-theory-ii-3.2.9hodge-theory-ii-3.2.9.xml3.2.9hodge-theory-ii-3.2
The recurrent filtration F on {}_WE_2^{pq} is q-opposite to its complex conjugate.
We prove by induction on r that:
501Lemmahodge-theory-ii-3.2.10hodge-theory-ii-3.2.10.xml3.2.10hodge-theory-ii-3.2
For r \geqslant0, the differentials d_r of the spectral sequence {}_WE are strictly compatible with the recurrent filtration F.
For r \geqslant2, they are zero.
500Proofhodge-theory-ii-3.2.10
For r=0 (resp. r=1) we apply and (iii) of (resp. ).
For r \geqslant2, it suffices to prove that d_r=0.
By induction, and following , we can assume that, on the pages {}_WE^s (for s \geqslant r+1), we have F_d=F_r=F_{d^*}, and that {}_WE_r={}_WE_2.
By (i) of , d_r is thus compatible with the filtration F_r.
On {}_WE_r^{pq}={}_WE_2^{pq}, the filtration F_r is q-opposite to its complex conjugate.
The morphism
d_r \colon {}_WE_r^{pq} \to {}_WE_r^{p+r,q-r+1}
thus satisfies, for r-1>0,
\begin {aligned} d_r({}_WE_r^{pq}) &= d_r \left ( \sum _{a+b=q} \Big ( F^a({}_WE_r^{pq}) \cap \bar {F}^b({}_WE_r^{pq}) \Big ) \right ) \\ & \subset \sum _{a+b=q} \Big ( F^a({}_WE_r^{p+r,q-r+1}) \cap \bar {F}^b({}_WE_r^{p+r,q-r+1}) \Big ) \\ &= 0. \end {aligned}
This proves .
, using , proves (i) of .
By , the filtration on {}_WE_ \infty ^{pq} induced by the filtration F of \operatorname {H} ^{p+q}(X, \mathbb {C} ) is q-opposite to its complex conjugate.
Since q=-p+(p+q), this proves the first part of (iii) of 507hodge-theory-ii-3.2.11hodge-theory-ii-3.2.11.xml3.2.11hodge-theory-ii-3.2
We now prove (ii) and (iii) of , which will finish the proof.
Independence from the choice of K^ \bullet.
The filtrations F and W of \operatorname {H} ^ \bullet (X, \mathbb {C} ) are the abutments of the hypercohomology spectral sequences of \Omega _{ \bar {X}}^ \bullet \langle Y \rangle for the filtrations F and W.
These whole spectral sequences do not depend on the choice of K^ \bullet.
Functoriality.
Let f \colon X_1 \to X_2 be a morphism of schemes.
Suppose that we are given a morphism of smooth compactifications
505hodge-theory-ii-3.2.11.1hodge-theory-ii-3.2.11.1.xml3.2.11.1hodge-theory-ii-3.2.11 \begin {CD} X_1 @>f>> X_2 \\ @V{j_1}VV @VV{j_2}V \\ \bar {X}_1 @>>{ \bar {f}}> \bar {X}_2 \end {CD} \tag{3.2.11.1}
with the Y_i= \bar {X}_i \setminus X_i normal crossing divisors.
The canonical morphism (cf. ) from \bar {f}^* \Omega _{ \bar {X}_2}^ \bullet \langle Y_2 \rangle to \Omega _{ \bar {X}_1}^ \bullet \langle Y_1 \rangle is then a morphism of bifiltered complexes;
on hypercohomology, it induces a morphism compatible with F and W, and thus f^* \colon \operatorname {H} ^n(X_2, \mathbb {Z} ) \to \operatorname {H} ^n(X_1, \mathbb {Z} ) is a morphism of mixed Hodge structures, for the structures defined by the compactifications \bar {X}_i.
Independence from the choice of compactification.
With the notation of (B), if f is an isomorphism, then f^* is a bijective morphism of mixed Hodge structures, and thus an isomorphism ().
If \bar {X}_1 and \bar {X}_2 are two smooth compactifications of X, with Y_i= \bar {X}_i \setminus X smooth crossing divisors, then there exists a third smooth compactification \bar {X}, with Y= \bar {X} \setminus X a smooth crossing divisor, that fits into a commutative diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& X \ar [dl,hook] \ar [d,hook] \ar [dr,hook]
\\ \bar {X}_1
& \bar {X} \ar [l] \ar [r]
& \bar {X}_2
\end {tikzcd}
Namely, we can take \bar {X} to be a resolution of singularities of the closure of the diagonal image of X in \bar {X}_1 \times \bar {X}_2.
The identity map from \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_1 to \operatorname {H} ^n(X, \mathbb {Z} ) endowed with the mixed Hodge structure defined by \bar {X}_2 is thus the composite of two isomorphisms.
To finish the proof of (ii) and (iii), we note that every morphism f fits into a diagram of the form in : we choose compactifications \bar {X}'_1 and \bar {X}_2 of X_1 and X_2, then we take \bar {X}_1 to be a resolution of singularities of the closure of the image of X_1 in \bar {X}'_1 \times \bar {X}_2.
508Definitionhodge-theory-ii-3.2.12hodge-theory-ii-3.2.12.xml3.2.12hodge-theory-ii-3.2
The mixed Hodge structure of the cohomology of a smooth separated algebraic variety is the mixed Hodge structure from (iii) of .
515Corollaryhodge-theory-ii-3.2.13hodge-theory-ii-3.2.13.xml3.2.13hodge-theory-ii-3.2
With the above notation:
The spectral sequence in degenerates at E_2, i.e. the Leray spectral sequence for the inclusion j \colon X^* \hookrightarrow X degenerates at E_3 (i.e. E_3=E_ \infty).
The spectral sequence in
{}_FE_1^{pq} = \operatorname {H} ^q( \bar {X}, \Omega _{ \bar {X}}^p \langle Y \rangle ) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {C} )
degenerates at E_1.
The spectral sequence defined by the sheaf \Omega _X^p \langle Y \rangle, endowed with the filtration W
\begin {aligned} E_1^{-n,k+n} &= \operatorname {H} ^k( \bar {X}, \operatorname {Gr} _n^W( \Omega _X^p \langle Y \rangle )) \\ & \approx \operatorname {H} ^k( \widetilde {Y}^n, \Omega _{ \widetilde {Y}^n}^{p-n}( \varepsilon ^n)) \Rightarrow \operatorname {H} ^k( \bar {X}, \Omega _X^p \langle Y \rangle ) \end {aligned}
degenerates at E_2.
514Proofhodge-theory-ii-3.2.13
Claim (i) is proven in .
Consider the four spectral sequences in , with respect to the bifiltered complex \Omega _{ \bar {X}}^ \bullet \langle Y \rangle.
By (i), we have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum _{p,q} \dim {}_WE_2^{p,q}.
The {}_WE_1^{n,k-n} pages are, for fixed n, the abutments of a spectral sequence that degenerates at E_1 () with initial pages E_1^{p,n,k-p-n} (with the notation of ) the initial pages of the spectral sequence in (iii).
Since {}_Wd_1 is strictly compatible with the filtration F which is the abutment of this spectral sequence (), and is () the abutment of a morphism between these spectral sequences, which starts with the differentials from (iii), we have
\operatorname {Gr} _F^p({}_WE_2^{n,k-n}) \approx \operatorname {H} ^ \bullet (E_1^{p,n-1,k-n-p} \to E_1^{p,n,k-n-p} \to E_1^{p,n+1,k-n-p})
and \operatorname {Gr} _F^ \bullet ({}_WE_2^{ \bullet \bullet }) is the sum of the pages E_2^{ \bullet \bullet \bullet } of the spectral sequences in (iii).
We thus have
513Equationhodge-theory-ii-3.2.13.1hodge-theory-ii-3.2.13.1.xml3.2.13.1hodge-theory-ii-3.2.13 \sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) = \sum \dim E_2^{ \bullet \bullet \bullet }. \tag{3.2.13.1}
We also have
\sum _n \dim \operatorname {H} ^n(X, \mathbb {C} ) \leqslant \dim {}_FE_1^{ \bullet \bullet }
with equality if and only if the spectral sequence in (ii) degenerates at E_1, and
\sum \dim {}_FE_1^{ \bullet \bullet } \leqslant \sum \dim E_2^{ \bullet \bullet \bullet }
with equality if and only if the spectral sequences in (iii) degenerate at E_2.
Comparing with , we obtain .
517Corollaryhodge-theory-ii-3.2.14hodge-theory-ii-3.2.14.xml3.2.14hodge-theory-ii-3.2
Let \omega be a meromorphic differential p-form on \bar {X} that is holomorphic on X and presents at worst logarithmic poles along Y.
Then the restriction \omega |X of \omega to X is closed, and if the cohomology class in \operatorname {H} ^p(X, \mathbb {C} ) defined by \omega is zero then \omega =0.
516Proofhodge-theory-ii-3.2.14
This is the particular case of (ii) of where {}_FE_1^{p0}={}_FE_ \infty ^{p0}.
522Corollaryhodge-theory-ii-3.2.15hodge-theory-ii-3.2.15.xml3.2.15hodge-theory-ii-3.2
If X is a smooth complete algebraic variety, then the mixed Hodge structure on \operatorname {H} ^n(X, \mathbb {Z} ) is the classical Hodge structure of weight n.
The Hodge numbers h^{pq} of the mixed Hodge structure of \operatorname {H} ^n(X, \mathbb {Z} ) (with X algebraic and smooth) can only be zero for p \leqslant n, q \leqslant n, and p+q \geqslant n
521Proofhodge-theory-ii-3.2.15
Claim (i) is clear;
to prove (ii), note that, if Y is the union of smooth divisors, then the rational Hodge structure \operatorname {Gr} _k^W( \operatorname {H} ^n(X, \mathbb {Q} )) is the quotient of a sub-object of a Hodge structure of weight n+k, namely
\operatorname {H} ^{n-k}( \widetilde {Y}^n, \mathbb {Q} ) \otimes \mathbb {Q} (-k).
523hodge-theory-ii-3.2.16hodge-theory-ii-3.2.16.xml3.2.16hodge-theory-ii-3.2
Let X be a smooth separated scheme.
We know that X admits smooth compactifications \bar {X} and that the subgroup of \operatorname {H} ^n(X, \mathbb {Z} ) given by the image of \operatorname {H} ^n( \bar {X}, \mathbb {Z} ) is independent of the choice of \bar {X} (cf. [G1968, (9.1) to (9.4)]).
525Corollaryhodge-theory-ii-3.2.17hodge-theory-ii-3.2.17.xml3.2.17hodge-theory-ii-3.2
Under the hypotheses of , the image of \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) in \operatorname {H} ^n(X, \mathbb {Q} ) is W_n( \operatorname {H} ^n(X, \mathbb {Q} )) (where W denotes the weight filtration ()).
524Proofhodge-theory-ii-3.2.17
We can suppose that \bar {X} \setminus X is a normal crossing divisor.
The claim then follows from the fact that W[-n] is the abutment of the Leray spectral sequence for the inclusion j \colon X \hookrightarrow \bar {X}.
527Corollaryhodge-theory-ii-3.2.18hodge-theory-ii-3.2.18.xml3.2.18hodge-theory-ii-3.2
Let f be a morphism from a smooth proper scheme Y to a smooth scheme X, with X admitting a smooth compactification j \colon X \hookrightarrow \bar {X}.
Then the groups \operatorname {H} ^n(X, \mathbb {Q} ) and \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) have the same image in \operatorname {H} ^n(Y, \mathbb {Q} ).
526Proofhodge-theory-ii-3.2.18
Since f^* and (jf)^* are strictly compatible with the weight filtration ((iii) of ), it suffices to prove that \operatorname {Gr} ^W(f^*) and \operatorname {Gr} ^W(f^*j^*) have the same image in \operatorname {Gr} ^W( \operatorname {H} ^n(Y, \mathbb {Q} )).
By and , \operatorname {Gr} _n^W(j^*) is an isomorphism, while \operatorname {Gr} _m^W(f^*)=0 for m \neq n since \operatorname {Gr} _m^W( \operatorname {H} ^n(Y, \mathbb {Q} ))=0 for m \neq n.
528Remarkhodge-theory-ii-3.2.19hodge-theory-ii-3.2.19.xml3.2.19hodge-theory-ii-3.2
By and , under the hypotheses of , the Hodge filtration on \operatorname {H} ^n(X, \mathbb {C} ) is the abutment of the hypercohomology spectral sequence of the complex j_*^m \Omega _{X^*}^ \bullet endowed with the filtration given by the pole-order filtration (): this spectral sequence coincides with .
1705hodge-theory-ii-2.3hodge-theory-ii-2.3.xmlHodge Theory II › Hodge structures › Mixed structures2.3hodge-theory-ii-21218Definitionhodge-theory-ii-2.3.1hodge-theory-ii-2.3.1.xml2.3.1hodge-theory-ii-2.3
A mixed Hodge structure H consists of
A \mathbb {Z}-module H_ \mathbb {Z} of finite type, called the "integral lattice";
A finite increasing filtration W_n on H_ \mathbb {Q} = \mathbb {Q} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the weight filtration;
A finite filtration F^p on H_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {Z} H_ \mathbb {Z}, called the Hodge filtration.
We demand that, on H_ \mathbb {C}, the filtration W_ \mathbb {C} induced by extension of scalars of W, the filtration F, and the complex conjugate \bar {F} form a system (W_ \mathbb {C} ,F, \bar {F}) of three opposite filtrations ( and ).
1219hodge-theory-ii-2.3.2hodge-theory-ii-2.3.2.xml2.3.2hodge-theory-ii-2.3
We also denote by W the filtration of H_ \mathbb {Z} given by the inverse image of the filtration W on H_ \mathbb {Q}.
The axiom of mixed Hodge structures implies that, for each n, the filtration F induces on \mathbb {C} \otimes _ \mathbb {Z} \operatorname {Gr} _n^W(H_ \mathbb {Z} ) a filtration that is n-opposite to its complex conjugate.
By , \operatorname {Gr} _n^W(H_ \mathbb {Z} ) is endowed with a Hodge structure of weight n, with Hodge filtration induced by F.
1220Examplehodge-theory-ii-2.3.3hodge-theory-ii-2.3.3.xml2.3.3hodge-theory-ii-2.3
If H is a Hodge structure of weight n, then we define a mixed Hodge structure with the same integral lattice and the same Hodge filtration by setting
W_i(H_ \mathbb {Q} ) = \begin {cases} 0 & \text {for } i<n \\ H_ \mathbb {Q} & \text {for } i \geqslant n. \end {cases} 1221hodge-theory-ii-2.3.4hodge-theory-ii-2.3.4.xml2.3.4hodge-theory-ii-2.3
A morphism f \colon H \to H' of Hodge structures is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} that is compatible with the filtrations W and F (and thus compatible with \bar {F}).
We immediately deduce from the following theorem.
1222Theoremhodge-theory-ii-2.3.5hodge-theory-ii-2.3.5.xml2.3.5hodge-theory-ii-2.3
The category of mixed Hodge structures is abelian.
The integral lattice of the kernel (resp. cokernel) of a morphism f \colon H \to H' is the kernel (resp. cokernel) K of f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z}, with K \otimes \mathbb {Q} and K \otimes \mathbb {C} being endowed with the induced filtrations (resp. quotient filtrations) of the filtrations W and F of H_ \mathbb {Q} and H_ \mathbb {C} (resp. of H'_ \mathbb {Q} and H'_ \mathbb {C}).
Every morphism f \colon H \to H' is strictly compatible with the filtration W of H_ \mathbb {Q} and H'_ \mathbb {Q} and with the filtration F of H_ \mathbb {C} and H'_ \mathbb {C}.
It induces morphisms of Hodge \mathbb {Q}-structures
\operatorname {Gr} _n^w(f) \colon \operatorname {Gr} _n^W(H_ \mathbb {Q} ) \to \operatorname {Gr} _n^W(H'_ \mathbb {Q} ).
It also induces morphisms
\operatorname {Gr} _F^p(F) \colon \operatorname {Gr} _F^p(H_ \mathbb {C} ) \to \operatorname {Gr} _F^p(H'_ \mathbb {C} )
that are strictly compatible with the filtration induced by W_ \mathbb {C}.
The functor \operatorname {Gr} _n^W is an exact functor from the category of mixed Hodge structures to the category of Hodge \mathbb {Q}-structures of weight n.
The functor \operatorname {Gr} _F^p is an exact functor.
1223hodge-theory-ii-2.3.6hodge-theory-ii-2.3.6.xml2.3.6hodge-theory-ii-2.3
Let H be a mixed Hodge structure.
The W_n(H_ \mathbb {Z} ), endowed with the filtrations induced by W and F, then form mixed Hodge substructures W_n(H) of H.
The quotient W_n(H)/W_{n-1}(H) can be identified with \operatorname {Gr} _n^W(H_ \mathbb {Z} ) endowed with its mixed Hodge structure ( and ).
We denote this Hodge structure by \operatorname {Gr} _n^W(H).
1224hodge-theory-ii-2.3.7hodge-theory-ii-2.3.7.xml2.3.7hodge-theory-ii-2.3
We set
H^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _{p+q}^W(H_ \mathbb {C} ) = ( \operatorname {Gr} _{p+q}^W(H))^{p,q}.
The Hodge numbers of H are the integers
h^{p,q} = \dim _ \mathbb {C} H^{p,q}.
The Hodge number h^{pq} of H is thus the Hodge number h^{pq} of the Hodge structure \operatorname {Gr} _{p+q}^W(H)1225hodge-theory-ii-2.3.8hodge-theory-ii-2.3.8.xml2.3.8hodge-theory-ii-2.3
We define a mixed Hodge \mathbb {Q}-structure H to consist of a vector space H_ \mathbb {Q} of finite dimension over \mathbb {Q}, a finite increasing filtration W on H_ \mathbb {Q}, and a finite decreasing filtration F on H_ \mathbb {C}, with the filtrations W_ \mathbb {C}, F, and \bar {F} being opposite.
generalises trivially to this variation.
1706hodge-theory-ii-1.2hodge-theory-ii-1.2.xmlHodge Theory II › Filtrations › Opposite filtrations1.2hodge-theory-ii-1657hodge-theory-ii-1.2.1hodge-theory-ii-1.2.1.xml1.2.1hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with filtrations F and G.
By definition, \operatorname {Gr} _F^n(A) is a quotient of a sub-object of A and, as such, is endowed with a filtration induced by G .
Passing to the associated graded defines a bigraded object ( \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A))_{n,m \in \mathbb {Z} }.
By a lemma of Zassenhaus, \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) and \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) are canonically isomorphic: if we define the induced filtrations as quotient filtrations of the induced filtrations on a sub-object, then we have
\begin {aligned} \operatorname {Gr} _G^n \operatorname {Gr} _F^m(A) \cong (F^m(A) \cap G^n(A)) &/ ((F^{m+1}(A) \cap G^n(A)) + (F^m(A) \cap G^{n+1}(A))) \\ &= \\ \operatorname {Gr} _F^m \operatorname {Gr} _G^n(A) \cong (G^n(A) \cap F^m(A)) &/ ((G^{n+1}(A) \cap F^m(A)) + (G^n(A) \cap F^{m+1}(A))) \end {aligned} 658hodge-theory-ii-1.2.2hodge-theory-ii-1.2.2.xml1.2.2hodge-theory-ii-1.2
Let H be a third filtration of A.
It induces a filtration on \operatorname {Gr} _F(A), and thus on \operatorname {Gr} _G \operatorname {Gr} _F(A).
It also induces a filtration on \operatorname {Gr} _F \operatorname {Gr} _G(A).
We note that these filtrations do not in general correspond to one another under the isomorphism .
In the expression \operatorname {Gr} _H \operatorname {Gr} _G \operatorname {Gr} _F(A), G and H thus play a symmetric role, but not F and G.
659Definitionhodge-theory-ii-1.2.3hodge-theory-ii-1.2.3.xml1.2.3hodge-theory-ii-1.2
Two finite filtrations F and \bar {F} on A are said to be n-opposite if \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q \neq n.
666hodge-theory-ii-1.2.4hodge-theory-ii-1.2.4.xml1.2.4hodge-theory-ii-1.2
If A^{p,q} is a bigraded object of \mathscr {A} such that
A^{p,q}=0 except for a finite number of pairs (p,q), and
A^{p,q}=0 for p+q \neq n
then we define two n-opposite filtrations of A= \sum _{p,q}A^{p,q} by setting
663Equationhodge-theory-ii-1.2.4.1hodge-theory-ii-1.2.4.1.xml1.2.4.1hodge-theory-ii-1.2.4 F^p(A) = \sum _{p' \geqslant p} A^{p',q'} \tag{1.2.4.1}
664Equationhodge-theory-ii-1.2.4.2hodge-theory-ii-1.2.4.2.xml1.2.4.2hodge-theory-ii-1.2.4 \bar {F}^q(A) = \sum _{q' \geqslant q} A^{p',q'}. \tag{1.2.4.2}
We have
665Equationhodge-theory-ii-1.2.4.3hodge-theory-ii-1.2.4.3.xml1.2.4.3hodge-theory-ii-1.2.4 \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A) = A^{p,q}. \tag{1.2.4.3}
Conversely:
667Propositionhodge-theory-ii-1.2.5hodge-theory-ii-1.2.5.xml1.2.5hodge-theory-ii-1.2
Let F and \bar {F} be finite filtrations on A.
For F and \bar {F} to be n-opposite, it is necessary and sufficient that, for all p,q,
[p+q=n+1] \implies [F^p(A) \oplus \bar {F}^q(A) \xrightarrow { \sim } A].
If F and \bar {F} are n-opposite, and if we set
\begin {cases} A^{p,q} = 0 & \text {for }p+q \neq n \\ A^{p,q} = F^p(A) \cap \bar {F}^q(A) & \text {for }p+q=n \end {cases}
then A is the direct sum of the A^{p,q}, and F and \bar {F} come from the bigrading A^{p,q} of A by the procedure of .
608Proofhodge-theory-ii-1.2.5
The condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n implies that F^p \cap \bar {F}^q=(F^{p+1} \cap \bar {F}^q)+(F^p \cap \bar {F}^{q+1}) for p+q>n.
By hypothesis, F^p \cap \bar {F}^q is zero for large enough p+q;
by decreasing induction, we thus deduce that the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q>n is equivalent to the condition F^p(A) \cap \bar {F}^q(A)=0 for p+q>n.
Dually (, , ), the condition \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q(A)=0 for p+q<n is equivalent to the condition A=F^p(A)+ \bar {F}^q(A) for (1-p)+(1-q)>-n, i.e. for p+q \leqslant n+1, and the claim then follows.
If F and \bar {F} are n-opposite, then we can prove by decreasing induction on p that
604Equationhodge-theory-ii-1.2.5.1hodge-theory-ii-1.2.5.1.xml1.2.5.1hodge-theory-ii-1.2.5 \bigoplus _{p' \geqslant p} A^{p',q'} \xrightarrow { \sim } F^p(A). \tag{1.2.5.1}
For F^p(A)=0, the claim is evident.
The decomposition A=F^{p+1}(A) \oplus \bar {F}^{n-p}(A) induces on F^p(A) \supset F^{p+1}(A) a decomposition
F^p(A) = F^{p+1}(A) \oplus (F^p(A) \cap \bar {F}^{n-p}(A))
and we conclude by induction.
For p small enough, we have F^p(A)=A.
By , the A^{p,q} thus form a bigrading of A, and F satisfies .
The fact that \bar {F} satisfies then follows by symmetry.
668hodge-theory-ii-1.2.6hodge-theory-ii-1.2.6.xml1.2.6hodge-theory-ii-1.2
The constructions and establish equivalences of categories that are quasi-inverse to one another between objects of \mathscr {A} endowed with two finite n-opposite filtrations and bigraded objects of \mathscr {A} of the type considered in .
669Definitionhodge-theory-ii-1.2.7hodge-theory-ii-1.2.7.xml1.2.7hodge-theory-ii-1.2
Three finite filtrations W, F, and \bar {F} on A are said to be opposite if
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A) = 0
for p+q+n \neq0.
This condition is symmetric in F and \bar {F}.
It implies that F and \bar {F} induce on W^n(A)/W^{n+1}(A) two (-n)-opposite filtrations.
We set
A^{p,q} = \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _F^{-p-q}(A)
whence decompositions ,
W^n(A)/W^{n+1}(A) = \bigoplus _{p+q=-n} A^{p,q} \tag{1.2.7.1}
which makes \operatorname {Gr} _W(A) into a bigraded object.
675Lemmahodge-theory-ii-1.2.8hodge-theory-ii-1.2.8.xml1.2.8hodge-theory-ii-1.2
Let W, F, and \bar {F} be three finite opposite filtrations, and \sigma a sequence (p_i,q_i)_{i \geqslant0 } pairs of integers satisfying
p_i \leqslant p_j and q_i \leqslant q_j for i \geqslant j, and
p_i+q_i=p_0+q_0-i+1 for i>0.
Set p=p_0, q=q_0, n=-p-q, and
A_ \sigma = \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap F^{p_i}(A)) \right ) \cap \left ( \sum _{0 \leqslant i}(W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \right ).
Then the projection from W^n(A) to \operatorname {Gr} _W^n(A) induces an isomorphism
A_ \sigma \xrightarrow { \sim } A^{p,q} \subset \operatorname {Gr} _W^n(A).
674Proofhodge-theory-ii-1.2.8
We will prove by induction on k the following claim:
The projection from W^n(A)/W^{n+k} to \operatorname {Gr} _W^n(A) induces an isomorphism from
\begin {aligned} \Bigg [ & \left ( \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) + W^{n+k}(A) \right ) \\ \cap & \left ( \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) + W^{n+k}(A) \right ) \Bigg ] / W^{n+k}(A) \end {aligned} \tag{$*_k$}
to A^{p,q} \subset \operatorname {Gr} _W^n(A).
For k=1, this is exactly the definition of A^{p,q}.
By (i) of we have
673Equationhodge-theory-ii-1.2.8.1hodge-theory-ii-1.2.8.1.xml1.2.8.1hodge-theory-ii-1.2.8 F^{p_k}( \operatorname {Gr} _W^{n+k}(A)) \oplus \bar {F}^{q_k}( \operatorname {Gr} _W^{n+k}(A)) \xrightarrow { \sim } \operatorname {Gr} _W^{n+k}(A). \tag{1.2.8.1}
Set
\begin {aligned} B &= \sum _{i<k} (W^{n+i}(A) \cap F^{p_i}(A)) \\ C &= \sum _{i<k} (W^{n+i}(A) \cap \bar {F}^{q_i}(A)) \\ B' &= (W^{n+k}(A) \cap F^{p_k}(A)) + W^{n+k+1}(A) \\ C' &= (W^{n+k}(A) \cap \bar {F}^{q_k}(A)) + W^{n+k+1}(A) \\ D &= W^{n+k}(A) \\ E &= W^{n+k+1}(A). \end {aligned}
can then be written as
\begin {aligned} B'+C' &= D \\ B' \cap C' &= E. \end {aligned}
We also have, since p_k \leqslant p_i (for i \leqslant k),
B \cap D \subset F^{p_k}(A) \cap W^{n+k}(A) \subset B'
and, since q_k \leqslant q_i (for i \leqslant k),
C \cap D \subset \bar {F}^{q_k}(A) \cap W^{n+k}(A) \subset C'.
The claim (*_{k+1}) and then follows from (*_k) and the following lemma.
677Lemmahodge-theory-ii-1.2.9hodge-theory-ii-1.2.9.xml1.2.9hodge-theory-ii-1.2
Let B, C, B', C', D, and E be sub-objects of A.
Suppose that
\begin {gathered} B'+C' = D \qquad B' \cap C' = E \\ B \cap D \subset B' \qquad C \cap D \subset C'. \end {gathered}
Then
((B+B') \cap (C+C'))/E \xrightarrow { \sim } ((B+D) \cap (C+D))/D.
676Proofhodge-theory-ii-1.2.9
To prove surjectivity, we write
\begin {aligned} ((B+B') \cap (C+C'))+D &= (((B+B') \cap (C+C'))+B')+C' \\ &= ((B+B') \cap (C+C'+B'))+C' \\ &= (B+B'+C') \cap (C+C'+B') \\ &= (B+D) \cap (C+D). \end {aligned}
To prove injectivity, we write
(B+B') \cap (C+C') \cap D = ((B+B') \cap D) \cap ((C+C') \cap D).
Since B' \subset D, we have
\begin {aligned} (B+B') \cap D &= (B \cap D)+B' \\ &= B' \end {aligned}
and similarly
(C+C') \cap D = C'
and
\begin {aligned} (B+B') \cap (C+C') \cap D &= B' \cap C' \\ &= E. \end {aligned}
This finishes the proof of , noting that is equivalent to (*_k) for large k.
683Theoremhodge-theory-ii-1.2.10hodge-theory-ii-1.2.10.xml1.2.10hodge-theory-ii-1.2
Let \mathscr {A} be an abelian category, and provisionally denote by \mathscr {A}' the category of objects of \mathscr {A} endowed with three opposite filtrations W, F, and \bar {F}.
The morphisms in \mathscr {A}' are the morphisms of \mathscr {A} that are compatible with the three filtrations.
\mathscr {A}' is an abelian category.
The kernel (resp. cokernel) of an arrow f \colon A \to B in \mathscr {A}' is the kernel (resp. cokernel) of f in \mathscr {A}, endowed with the filtrations induced by those of A (resp. the quotients of those of B).
Every morphism f \colon A \to B in \mathscr {A}' is strictly compatible with the filtrations W, F, and \bar {F};
the morphism \operatorname {Gr} _W(f) is compatible with the bigradings of \operatorname {Gr} _W(A) and \operatorname {Gr} _W(B);
the morphisms \operatorname {Gr} _F(f) and \operatorname {Gr} _{ \bar {F}}(f) are strictly compatible with the filtration induced by W.
The "forget the filtrations" functors, \operatorname {Gr} _W, \operatorname {Gr} _F, and \operatorname {Gr} _{ \bar {F}}, and
\begin {gathered} \operatorname {Gr} _W \operatorname {Gr} _F \simeq \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _F \operatorname {Gr} _W \\ \simeq \operatorname {Gr} _{ \bar {F}} \operatorname {Gr} _W \simeq \operatorname {Gr} _W \operatorname {Gr} _{ \bar {F}} \end {gathered}
from \mathscr {A}' to \mathscr {A} are exact.
Denote by \sigma _0(p,q) and \sigma _1(p,q) the sequences
\begin {aligned} \sigma _0(p,q) &= (p,q), (p,q), (p,q-1), (p,q-2), (p,q-3), \ldots \\ \sigma _0(p,q) &= (p,q), (p,q), (p-1,q), (p-2,q), (p-3,q), \ldots \end {aligned}
and, with the notation of , set
A_i^{p,q} = A_{ \sigma _i(p,q)} \qquad \text {for } i=0,1.
If f \colon A \to B is compatible with W, F, and \bar {F}, then we have
684Equationhodge-theory-ii-1.2.10.1hodge-theory-ii-1.2.10.1.xml1.2.10.1hodge-theory-ii-1.2 f(A_i^{p,q}) \subset B_i^{p,q} \qquad \text {for }i=0,1. \tag{1.2.10.1}
Claim (iii) then follows from the following lemma:
690Lemmahodge-theory-ii-1.2.11hodge-theory-ii-1.2.11.xml1.2.11hodge-theory-ii-1.2
The A_i^{p,q} give a bigrading of A.
We have
685Equationhodge-theory-ii-1.2.11.1hodge-theory-ii-1.2.11.1.xml1.2.11.1hodge-theory-ii-1.2.11 W^n(A) = \sum _{n+p+q \leqslant 0} A_i^{p,q} \qquad \text {for }i=0,1 \tag{1.2.11.1}
686Equationhodge-theory-ii-1.2.11.2hodge-theory-ii-1.2.11.2.xml1.2.11.2hodge-theory-ii-1.2.11 F^p(A) = \sum _{p' \geqslant p} A_0^{p',q'} \tag{1.2.11.2}
687Equationhodge-theory-ii-1.2.11.3hodge-theory-ii-1.2.11.3.xml1.2.11.3hodge-theory-ii-1.2.11 \bar {F}^q(A) = \sum _{q' \geqslant q} A_1^{p',q'}. \tag{1.2.11.3}
689Proofhodge-theory-ii-1.2.11
By symmetry, it suffices to prove the claims concerning i=0.
Set A_0= \bigoplus A_0^{p,q} and define filtrations W and F on A_0 by the equations of .
The canonical map i from A_0 to A is compatible with the filtrations W and F.
Furthermore, by , \operatorname {Gr} _W(i) is an isomorphism, and induces isomorphisms of graded objects
688Equationhodge-theory-ii-1.2.11.4hodge-theory-ii-1.2.11.4.xml1.2.11.4hodge-theory-ii-1.2.11 \sum _{p+q=n} A_0^{p,q} \xrightarrow { \sim } \operatorname {Gr} _W^{-n}(A) = \sum _{p+q=n} A^{p,q}. \tag{1.2.11.4}
The morphism i is thus an isomorphism, and the A_0^{p,q} give a bigrading of A.
then says that \operatorname {Gr} _W(i) is an isomorphism.
By , \operatorname {Gr} _F \operatorname {Gr} _W(i) is an isomorphism, and thus so too are \operatorname {Gr} _W \operatorname {Gr} _F(i) and \operatorname {Gr} _F(i).
Equation then follows.
691hodge-theory-ii-1.2.12hodge-theory-ii-1.2.12.xml1.2.12hodge-theory-ii-1.2
We now prove .
Let f \colon A \to B in \mathscr {A}' and endow K= \operatorname {Ker} (f) with the filtrations induced by those of A.
By , \operatorname {Gr} _W(K) \hookrightarrow \operatorname {Gr} _W(A);
furthermore, the filtration F (resp. \bar {F}) on K induces on \operatorname {Gr} _W(K) the inverse image filtration of the filtration F on \operatorname {Gr} _W(A).
The sub-object \operatorname {Gr} _W(K) of \operatorname {Gr} _W(A) is then compatible with the bigrading of \operatorname {Gr} _W(A):
\operatorname {Gr} _W(K) = \bigoplus _{p,q}( \operatorname {Gr} _W(K) \cap A^{p,q}).
We thus deduce that
\operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(K) \hookrightarrow \operatorname {Gr} _F^p \operatorname {Gr} _{ \bar {F}}^q \operatorname {Gr} _W^n(A);
the filtrations of W, F, and \bar {F} on K are thus opposite, and K is a kernel of f in \mathscr {A}'.
This, combined with the dual result, proves (ii).
If f is an arrow of \mathscr {A}', then the canonical morphism from \operatorname {Coim} (f) to \operatorname {Im} (f) is an isomorphism in \mathscr {A};
by (iii), it is also an isomorphism in \mathscr {A}', which is thus abelian.
The "forget the filtrations" functor is exact by (ii).
The exactness of the other functors in (iv) follows immediately from (ii), (iii), and (i) or (ii) of .
692hodge-theory-ii-1.2.13hodge-theory-ii-1.2.13.xml1.2.13hodge-theory-ii-1.2
Let A be an object of \mathscr {A} endowed with a finite increasing filtration W_ \bullet, and two finite decreasing filtrations F and \bar {F}.
The construction associates to W_ \bullet a decreasing filtration W^ \bullet.
We say that the filtrations W_ \bullet, F, and \bar {F} are opposite if the filtrations W^ \bullet, F, and \bar {F} are, i.e. if, for all n, the filtrations induced by F and \bar {F} on
\operatorname {Gr} _n^W(A) = W_n(A)/W_{n-1}(A)
are n-opposite.
translates trivially to this variation.
1707hodge-theory-ii-2.1hodge-theory-ii-2.1.xmlHodge Theory II › Hodge structures › Pure structures2.1hodge-theory-ii-2773hodge-theory-ii-2.1.1hodge-theory-ii-2.1.1.xml2.1.1hodge-theory-ii-2.1
In all the following, we denote by \mathbb {C} an algebraic closure of \mathbb {R}, and we do not suppose to have chosen a root i of the equation x^2+1=0.
The theory will be invariant under complex conjugation (cf. [hodge-theory-ii-2.1.14 (?)]).
774hodge-theory-ii-2.1.2hodge-theory-ii-2.1.2.xml2.1.2hodge-theory-ii-2.1
We denote by S the real algebraic group \mathbb {C} ^*, given by restriction of scalars à la Weil from \mathbb {C} to \mathbb {R} of the group \mathbb {G}_ \mathrm {m}:
\begin {aligned} S &= \prod _{ \mathbb {C} {/} \mathbb {R} } \mathbb {G}_ \mathrm {m} \\ S( \mathbb {R} ) &= \mathbb {C} ^*. \end {aligned}
The group S is a torus, i.e. it is connected and of multiplicative type.
It is thus described by the free abelian group of finite type
X(S) = \operatorname {Hom} (S_ \mathbb {C} , \mathbb {G}_ \mathrm {m} ) = \underline { \operatorname {Hom} } (S, \mathbb {G}_ \mathrm {m} )( \mathbb {C} )
of its complex characters, endowed with the action of \operatorname {Gal} ( \mathbb {C} {/} \mathbb {R} )= \mathbb {Z} {/}(2).
The group X(S) has generators z and \bar {z}, which induce (respectively) the identity and complex conjugation:
\mathbb {C} ^* = S( \mathbb {R} ) \to S( \mathbb {C} ) \to \mathbb {G}_ \mathrm {m} ( \mathbb {C} ) = \mathbb {C} ^*.
Complex conjugation exchanges z and \bar {z}.
780hodge-theory-ii-2.1.3hodge-theory-ii-2.1.3.xml2.1.3hodge-theory-ii-2.1
We have a canonical map
775Equationhodge-theory-ii-2.1.3.1hodge-theory-ii-2.1.3.1.xml2.1.3.1hodge-theory-ii-2.1.3 w \colon \mathbb {G}_ \mathrm {m} \to S \tag{2.1.3.1}
that, on the real points, induces the inclusion of \mathbb {R} ^* into \mathbb {C} ^*.
We have
776Equationhodge-theory-ii-2.1.3.2hodge-theory-ii-2.1.3.2.xml2.1.3.2hodge-theory-ii-2.1.3 zw = \bar {z}w = \mathrm {Id} . \tag{2.1.3.2}
We also have a map
777Equationhodge-theory-ii-2.1.3.3hodge-theory-ii-2.1.3.3.xml2.1.3.3hodge-theory-ii-2.1.3 N \colon S \to \mathbb {G}_ \mathrm {m} \tag{2.1.3.3}
that on the real points can be identified with the norm N_{ \mathbb {C} {/} \mathbb {R} } \colon \mathbb {C} ^* \to \mathbb {R} ^*.
We have
778Equationhodge-theory-ii-2.1.3.4hodge-theory-ii-2.1.3.4.xml2.1.3.4hodge-theory-ii-2.1.3 N = z \bar {z} \tag{2.1.3.4}
779Equationhodge-theory-ii-2.1.3.5hodge-theory-ii-2.1.3.5.xml2.1.3.5hodge-theory-ii-2.1.3 N \circ w = (x \mapsto x^2). \tag{2.1.3.5} 781Definitionhodge-theory-ii-2.1.4hodge-theory-ii-2.1.4.xml2.1.4hodge-theory-ii-2.1
A real Hodge structure is a real vector space V of finite dimension endowed with an action of the real algebraic group S.
783hodge-theory-ii-2.1.5hodge-theory-ii-2.1.5.xml2.1.5hodge-theory-ii-2.1
By the general theory of groups of multiplicative type, giving a real Hodge structure on V is equivalent to giving a bigrading V^{p,q} of V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that satisfies \overline {V^{pq}}=V^{qp}.
The action of S and the bigrading are mutually determined via the condition:
782hodge-theory-ii-2.1.5.1hodge-theory-ii-2.1.5.1.xml2.1.5.1hodge-theory-ii-2.1.5
On V^{pq}, S acts by multiplication by z^p \bar {z}^q.
784hodge-theory-ii-2.1.6hodge-theory-ii-2.1.6.xml2.1.6hodge-theory-ii-2.1
Let ( \mathbb {C} , \times ) be the multiplicative monoid, and let
\bar {S} = \prod _{ \mathbb {C} {/} \mathbb {R} } ( \mathbb {C} , \times ).
If V is a real vector space, then we can show that it is equivalent to either give an action of \bar {S} on V or to give a bigrading V^{pq} on V such that \overline {V^{pq}}=V^{qp} and V^{pq}=0 for p<0 or q<0.
785hodge-theory-ii-2.1.7hodge-theory-ii-2.1.7.xml2.1.7hodge-theory-ii-2.1
Let V be a real Hodge structure, defined by a representation \sigma of S and a bigrading V^{pq}.
The grading of V_ \mathbb {C} by the V_ \mathbb {C} ^n= \sum _{p+q=n}V^{pq} is then defined over \mathbb {R}.
We call this the weight grading.
On V^n=V \cap V_ \mathbb {C} ^n, the representation \sigma w of \mathbb {G}_ \mathrm {m} is multiplication by x^n.
We say that V is of weight n if V^{pq}=0 for p+q \neq n, i.e. if \sigma w is multiplication by x^n.
786hodge-theory-ii-2.1.8hodge-theory-ii-2.1.8.xml2.1.8hodge-theory-ii-2.1
Let V be a real Hodge structure.
The Hodge filtration on V_ \mathbb {C} is defined by
F^p(V_ \mathbb {C} ) = \sum _{p' \geqslant p} V^{p'q'}.
By , we have:
790Propositionhodge-theory-ii-2.1.9hodge-theory-ii-2.1.9.xml2.1.9hodge-theory-ii-2.1
Let n be an integer.
The construction establishes an equivalence of categories between:
the category of real Hodge structures of weight n;
the category of pairs consisting of a real vector space V of finite dimension and of a filtration F on V_ \mathbb {C} = \mathbb {C} \otimes _ \mathbb {R} V that is n-opposite to its complex conjugate \bar {F}.
794Definitionhodge-theory-ii-2.1.10hodge-theory-ii-2.1.10.xml2.1.10hodge-theory-ii-2.1
A Hodge structure H, of weight n, consists of
a \mathbb {Z}-module H_ \mathbb {Z} of finite type (the "integral lattice");
a real Hodge structure of weight n on H_ \mathbb {R} = \mathbb {R} \otimes _ \mathbb {Z} H_ \mathbb {Z}.
795hodge-theory-ii-2.1.11hodge-theory-ii-2.1.11.xml2.1.11hodge-theory-ii-2.1
A morphism f \colon H \to H' is a homomorphism f \colon H_ \mathbb {Z} \to H'_ \mathbb {Z} such that f_ \mathbb {R} \colon H_ \mathbb {R} \to H'_ \mathbb {R} is compatible with the action of S (i.e. such that f_ \mathbb {C} is compatible with the bigrading, or with the Hodge filtration).
Hodge structures of weight n form an abelian category.
If H is of weight n, and H' is of weight n', then we define a Hodge structure H \otimes H' of weight n+n' by the equations:
(H \otimes H')_ \mathbb {Z} =H_ \mathbb {Z} \otimes H'_ \mathbb {Z};
the action of S on (H \otimes H')_ \mathbb {R} =H_ \mathbb {R} \otimes H'_ \mathbb {R} is the tensor product of the actions of S on H_ \mathbb {R} and on H'_ \mathbb {R}.
The bigrading (resp. Hodge filtration) of (H \otimes H')_ \mathbb {C} =H_ \mathbb {C} \otimes H'_ \mathbb {C} is the tensor product of the bigradings (resp. Hodge filtrations (cf. )) of H_ \mathbb {C} and of H'_ \mathbb {C}.
We define in an analogous manner the Hodge structure \underline { \operatorname {Hom} } (H,H') (of weight n'-n), the Hodge structures \wedge ^p H (of weight pn), and the dual Hodge structure H^*.
The internal hom \underline { \operatorname {Hom} } above and the homomorphism group, are related by:
620Remarkhodge-theory-ii-2.1.11.1hodge-theory-ii-2.1.11.1.xml2.1.11.1hodge-theory-ii-2.1.11\operatorname {Hom} (H,H') is the subgroup of
\underline { \operatorname {Hom} } (H,H')_ \mathbb {Z} = \operatorname {Hom} _ \mathbb {Z} (H_ \mathbb {Z} ,H'_ \mathbb {Z} )
consisting of elements of degree (0,0).
The actions of S on H_ \mathbb {R}, H'_ \mathbb {R}, and \underline { \operatorname {Hom} } (H,H')_ \mathbb {R} = \operatorname {Hom} (H_ \mathbb {R} ,H'_ \mathbb {R} ) are related by
s(f(x)) = s(f)(s(x)).
This means that if f is of degree (0,0), i.e. invariant under S, then it commutes with the action of S.
796hodge-theory-ii-2.1.12hodge-theory-ii-2.1.12.xml2.1.12hodge-theory-ii-2.1
If A is a Noetherian subring of \mathbb {R}, then we define a Hodge A-structure of weight n to consist of an A-module H_A of finite type along with a real Hodge structure of weight n on H_ \mathbb {R} =H_A \otimes _A \mathbb {R}.
This definition is mostly used for A= \mathbb {Q}.
A Hodge A-structure consists of an A-module H_A of finite type along with a real Hodge structure on H_ \mathbb {R} =H_A \otimes _A \mathbb {R} such that the weight grading is defined over the field of fractions of A.
797Definitionhodge-theory-ii-2.1.13hodge-theory-ii-2.1.13.xml2.1.13hodge-theory-ii-2.1
The Tate Hodge structure \mathbb {Z} (1) is the Hodge structure of weight -2, of rank 1, of pure bidegree (-1,-1), with integral lattice 2 \pi i \mathbb {Z} \subset \mathbb {C}.
The action of S is thus given by multiplication with the inverse of the norm ().
For n \in \mathbb {Z}, we define \mathbb {Z} (n) as the n-th tensor power of \mathbb {Z} (1), so \mathbb {Z} (n) is the Hodge structure of weight -2n, of rank 1, of pure bidegree (-n,-n), with integral lattice (2 \pi i)^n \mathbb {Z} \subset \mathbb {C}.
THe action of S is multiplication by N(x)^{-n}.
798hodge-theory-ii-2.1.14hodge-theory-ii-2.1.14.xml2.1.14hodge-theory-ii-2.1
The choice in \mathbb {C} of a solution i of the equation x^2+1=0 determines, on each complex variety X of pure dimension n, an orientation \operatorname {or} _i(X).
Replace i by -i gives
\operatorname {or} _{-i}(X) = (-1)^n \operatorname {or} _i(X).
The choice of i also defines an element C of order 4 in S( \mathbb {R} ), given by the image of i under the isomorphism S( \mathbb {R} ) \simeq \mathbb {C} ^*.
Finally, it also defines an isomorphism between \mathbb {Z} and the integral lattice of \mathbb {Z} (n), given by multiplication by (2 \pi i)^n.
When i, an orientation of X, C, or an identification \mathbb {Z} \sim \mathbb {Z} (n)_ \mathbb {Z} appear in an equation, it is implicitly understood that they all follow from a single choice of the same i, and that by replacing i with -i we obtain an equivalent equation.
799Definitionhodge-theory-ii-2.1.15hodge-theory-ii-2.1.15.xml2.1.15hodge-theory-ii-2.1
A polarisation of a Hodge structure H of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {Z} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
800hodge-theory-ii-2.1.16hodge-theory-ii-2.1.16.xml2.1.16hodge-theory-ii-2.1
The real Tate Hodge structure is the real Hodge structure \mathbb {R} (1) underlying \mathbb {Z} (1).
We similarly define \mathbb {R} (n) as underlying \mathbb {Z} (1).
A polarisation of a real Hodge structure of weight n is a homomorphism
(x,y) \colon H \otimes H \to \mathbb {R} (-n)
such that the real bilinear form (2 \pi i)^n(x,Cy) on H_ \mathbb {R} is symmetric and positive definite.
A polarisation is entirely defined by the positive definite quadratic form (2 \pi i)^n(x,Cy) on H_ \mathbb {R}, imposed with only the condition of being invariant under the compact sub-torus of S given by the kernel of N.
We have
(x,y) = (Cx,Cy) = (y,C^2x) = (-1)^n(y,x).
The form (x,y) is thus symmetric or alternating, depending on the parity of n.
801hodge-theory-ii-2.1.17hodge-theory-ii-2.1.17.xml2.1.17hodge-theory-ii-2.1
The reader can generalise these definitions to Hodge A-structures of weight n ().
1708hodge-theory-ii-4.3hodge-theory-ii-4.3.xmlHodge Theory II › Applications and supplements › Supplement to [D1968]4.3hodge-theory-ii-41709hodge-theory-ii-4.1hodge-theory-ii-4.1.xmlHodge Theory II › Applications and supplements › The fixed set theorem4.1hodge-theory-ii-41495Theoremhodge-theory-ii-4.1.1hodge-theory-ii-4.1.1.xml4.1.1hodge-theory-ii-4.1
Let S be a smooth separated scheme, and f \colon X \to S a smooth proper morphism.
In rational cohomology, the Leray spectral sequence
E_2^{pq} = \operatorname {H} ^p(S, \mathrm {R} ^qf_* \mathbb {Q} ) \Rightarrow \operatorname {H} ^{p+q}(X, \mathbb {Q} )
degenerates (E_2=E_ \infty).
If \bar {X} is a non-singular compactification of X, then the canonical morphism
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^0(S, \mathrm {R} ^nf_* \mathbb {Q} )
is surjective.
266Proofhodge-theory-ii-4.1.1
For f smooth and projective, claim (i) is proven in D1968.
The proof in D1968 is rewritten in a more readable manner in G1970;
it does not use the smoothness of S.
Let us fix f, and prove that (i)\implies(ii).
We can reduce to the case where S is non-empty and connected.
If s \in S is a point, then the local system \mathrm {R} ^nf_* \mathbb {Q} is entirely described by its fibre ( \mathrm {R} ^nf_* \mathbb {Q} )_s at s and the action of the fundamental group \pi = \pi _1(S,s) on this fibre.
We have
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xrightarrow { \sim } [( \mathrm {R} ^n f_* \mathbb {Q} )_s]^ \pi .
If X_s=f^{-1}(s), then the composite arrow
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \to ( \mathrm {R} ^n f_* \mathbb {Q} ) \approx \operatorname {H} ^n(X_s, \mathbb {Q} )
is thus injective.
Consider the arrows
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \xrightarrow {a} \operatorname {H} ^n(X, \mathbb {Q} ) \xrightarrow {b} \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xhookrightarrow {c} \operatorname {H} ^n(X_s, \mathbb {Q} ).
The arrows cb and cba have the same image, by .
The arrow b is an "edge homomorphism" of the Leray spectral sequence, and is thus surjective by hypothesis.
Since c is injective, b and ba have the same image, and ba is surjective.
This proves (ii) for f projective.
From this we will deduce the general case.
We can again reduce to the case where S is non-empty and connected.
By Chow's lemma and the resolution of singularities, there exists a quasi-projective smooth scheme X' and a projective birational morphism p \colon X' \to X.
There thus exists (by Bertini, or Sard) a non-empty Zariski open S_1 of S such that X'_1=(fp)^{-1}(S_1) is smooth over S_1.
Finally, let X_1=f^{-1}(S_1), let \bar {X} be a smooth compactification of X, and let \bar {X}'_1 be a smooth compactification of X'_1 that dominates \bar {X}.
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
\bar {X}
&& \bar {X}'_1
\ar [ll,swap," \bar {p}"]
\\ X
\ar [u,hook]
\ar [d,swap,"f"]
& X_1
\ar [l,hook']
\ar [d,swap,"f_1"]
& X'_1
\ar [u,hook]
\ar [l,swap,"p"]
\ar [d,swap,"f'_1"]
\\ S
& S_1
\ar [l,hook',"i"]
& S_1
\ar [l,equals]
\end {tikzcd}
If s \in S_1, then the morphism i_* \colon \pi _1(S_1,s) \to \pi _1(S,s) is surjective, since the topological codimension of S \setminus S_1 is \geqslant2.
We thus have
\operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) \xrightarrow { \sim } \operatorname {H} ^0(S_1, \mathrm {R} ^n {f_1}_* \mathbb {Q} )
and it suffices to prove that
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^0(S_1, \mathrm {R} ^n {f_1}_* \mathbb {Q} )
is surjective.
The vertical arrows in the diagram
\begin {CD} \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) @>>> \operatorname {H} ^n(X_1, \mathbb {Q} ) @>>> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f_1}_* \mathbb {Q} ) \\ @VV{ \bar {p}^*}V @VV{p^*}V @VV{p^*}V \\ \operatorname {H} ^n( \bar {X}'_1, \mathbb {Q} ) @>>> \operatorname {H} ^n(X'_1, \mathbb {Q} ) @>>> \operatorname {H} ^0(S_1, \mathrm {R} {f'_1}_* \mathbb {Q} ) \end {CD}
admit left inverses given by the Gysin morphisms \bar {p}_!, p_!, and p_! (respectively), defined by Poincaré duality as the transposes of the arrows analogous to the above but in cohomology with proper support.
Furthermore, the diagram
\begin {CD} \operatorname {H} ^n( \bar {X}, \mathbb {Q} ) @>{u}>> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f_1}_* \mathbb {Q} ) \\ @A{ \bar {p}_!}AA @AA{p_!}A \\ \operatorname {H} ^n( \bar {X}', \mathbb {Q} ) @>>{v}> \operatorname {H} ^0(S_1, \mathrm {R} ^n{f'_1}_* \mathbb {Q} ) \end {CD}
commutes.
The arrow u is thus a direct factor of the arrow v.
Since v is surjective (because f'_1 is projective), so too is u.
We now prove (i) in the general case.
We can reduce to the case where f is of pure relative dimension n.
Let f \times f be the projection of X \times _S X to S.
Let \delta be the image of the cohomology class of the diagonal of X \times _S X in \operatorname {H} ^0(S, \mathrm {R} ^n(f \times f)_*, \mathbb {Q} ).
We have, by Künneth,
\mathrm {R} ^n(f \times f)_* \mathbb {Q} = \sum _{p+q=n} ( \mathrm {R} ^p f_* \mathbb {Q} \otimes \mathrm {R} ^q f_* \mathbb {Q} ).
We denote by \delta '_{pq} (for p+q=n) the components of \delta in this decomposition, and by \delta _{pq} the classes in \operatorname {H} ^n(X \times _S X) given by the images of the \delta '_{pq}.
The \delta _{pq} define, in the derived category D^+(S), homomorphisms
\delta _p \colon \mathrm {R} f_* \mathbb {Q} \to \mathrm {R} f_* \mathbb {Q}
such that \operatorname { \mathscr {H}} ^q( \delta _p) is zero for p \neq q, and is the identity for p=q.
By [D1968], we thus have
265Equationhodge-theory-ii-4.1.1.1hodge-theory-ii-4.1.1.1.xml4.1.1.1hodge-theory-ii-4.1.1 \mathrm {R} f_* \mathbb {Q} \approx \sum _p \mathrm {R} ^p f_* \mathbb {Q} [-p] \tag{4.1.1.1}
in D^+(S), and the Leray spectral sequence degenerates.
1496Corollaryhodge-theory-ii-4.1.2hodge-theory-ii-4.1.2.xml4.1.2hodge-theory-ii-4.1
Let f \colon X \to S be a proper smooth morphism, with S reduced, connected, and separated.
Let ( \mathrm {R} ^n f_* \mathbb {Q} )^0 be the largest constant local system on \mathrm {R} ^n f_* \mathbb {Q}, with fibre \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ).
Then, for each s \in S, ( \mathrm {R} ^n f_* \mathbb {Q} )_s^0 is a Hodge sub-structure of ( \mathrm {R} ^n f_* \mathbb {Q} ) \simeq \operatorname {H} ^n(X_s, \mathbb {Q} ), and the induced Hodge structure on \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ) is independent of s.
1492Proofhodge-theory-ii-4.1.2
Let s \in S, and let X_s=f^{-1}(s).
If S is smooth, and if \bar {X} is a smooth compactification of X, then, by , the subspace ( \mathrm {R} ^n f_* \mathbb {Q} )_s^0 of ( \mathrm {R} ^n f_* \mathbb {Q} )_s is the image of \operatorname {H} ^n( \bar {X}, \mathbb {Q} ).
Since the restriction map
\operatorname {H} ^n( \bar {X}, \mathbb {Q} ) \to \operatorname {H} ^n(X_s, \mathbb {Q} )
is a morphism of Hodge structures, its image is a Hodge sub-structure, and the induced Hodge structure on \operatorname {H} ^0(S, \mathrm {R} ^n f_* \mathbb {Q} ), given by the quotient of that on \operatorname {H} ^n( \bar {X}, \mathbb {Q} ), is independent of s.
In the general case, also implies that, if a is a global section of \mathrm {R} ^n f_* \mathbb {C}, then its components a^{p,q} of type (p,q), which are a priori only continuous sections of the complex bundle defined by \mathrm {R} ^n f_* \mathbb {C}, are in fact locally constant, i.e. are sections of \mathrm {R} ^n f_* \mathbb {C}.
This is the case because a^{p,q} is continuous, and further locally constant on the (dense) open subset of smooth points of S, by the above.
1497hodge-theory-ii-4.1.3hodge-theory-ii-4.1.3.xml4.1.3hodge-theory-ii-4.1
In the case where S is compact, a generalisation of is proven by analytic methods in [G1970].
As a corollary to , we note:
268hodge-theory-ii-4.1.3.1hodge-theory-ii-4.1.3.1.xml4.1.3.1hodge-theory-ii-4.1.3
(cf. [G1970]).
Let f \colon X \to S be a proper smooth morphism, as in .
If a global section a of \mathrm {R} ^n f_* \mathbb {C} is of Hodge type (p,q) at a point, then a is everywhere of type (p,q).
In particular, if n=2 and if a is, at some point s, the cohomology class of a divisor of X_s, then a is everywhere the cohomology class of a divisor, and, if S is smooth, is even defined by a divisor D on X.
269hodge-theory-ii-4.1.3.2hodge-theory-ii-4.1.3.2.xml4.1.3.2hodge-theory-ii-4.1.3
(cf. [G1966]).
Let S be a reduced connected scheme of finite type over \mathbb {C}, and let f_1 \colon X_1 \to S and f_2 \colon X_2 \to S be abelian schemes over S.
If a morphism u \colon \mathrm {R} ^1{f_2}_* \mathbb {Z} \to \mathrm {R} ^1{f_1}_* \mathbb {Z} comes from a morphism \widetilde {u}_s \colon (X_1)_s \to (X_2)_s of abelian varieties at some point s \in S, then u comes from a (unique) morphism of abelian schemes \widetilde {u} \colon X_1 \to X_2.
270hodge-theory-ii-4.1.3.3hodge-theory-ii-4.1.3.3.xml4.1.3.3hodge-theory-ii-4.1.3
(cf. [K1971]).
Let S be a smooth connected scheme, s \in S a point, f \colon X \to S a proper smooth morphism, and P a direct factor of the local system \mathrm {R} ^i f_* \mathbb {Q}, which is pointwise a Hodge sub-structure.
Then the following conditions are equivalent:
The Hodge structure of P is locally constant.
The representation P_s of \pi _1(S,s) factors through a finite quotient of \pi _1(S,s).
There exists a finite non-empty étale covering u \colon S' \to S such that u^*P is a constant family of Hodge structures.
1710hodge-theory-ii-4.2hodge-theory-ii-4.2.xmlHodge Theory II › Applications and supplements › The semi-simplicity theorem4.2hodge-theory-ii-41102hodge-theory-ii-4.2.1hodge-theory-ii-4.2.1.xml4.2.1hodge-theory-ii-4.2
Let S be a topological space.
A continuous family of Hodge structures on S consists of:
A local system H_ \mathbb {Z} on S of \mathbb {Z}-modules of finite type.
For every point s \in S, a Hodge structure on the fibre (H_ \mathbb {Z} ) that varies continuously in s.
A continuous family H of Hodge structures on S is said to be of weight n if the fibres H_s (for s \in S) are all of weight n.
We similarly define a continuous family of Hodge \mathbb {Q}-structures as a local system of \mathbb {Q}-vector spaces, endowed at each point with a Hodge \mathbb {Q}-structure, varying continuously.
A polarisation of a continuous family H of Hodge \mathbb {Q}-structures of weight n is a morphism of local systems from H_ \mathbb {Q} \otimes H_ \mathbb {Q} to the constant local system \mathbb {Q} (-n)_ \mathbb {Q}, which defines at each point s \in S a polarisation of H_s.
1107hodge-theory-ii-4.2.2hodge-theory-ii-4.2.2.xml4.2.2hodge-theory-ii-4.2
Suppose that S is connected, and let \mathcal {C} be a strictly full subcategory of the category of continuous families of Hodge \mathbb {Q}-structures on S.
We will have need of the following conditions:
1103Conditionhodge-theory-ii-4.2.2.1hodge-theory-ii-4.2.2.1.xml4.2.2.1hodge-theory-ii-4.2.2\mathcal {C} is stable under direct factors, direct sums, and tensor products;
the Tate constant families \mathbb {Q} (n) (for n \in \mathbb {Z}) are in \mathcal {C}.
1104Conditionhodge-theory-ii-4.2.2.2hodge-theory-ii-4.2.2.2.xml4.2.2.2hodge-theory-ii-4.2.2
Every homogeneous (of some weight n) Hodge structure in \mathcal {C} is polarisable.
1105Conditionhodge-theory-ii-4.2.2.3hodge-theory-ii-4.2.2.3.xml4.2.2.3hodge-theory-ii-4.2.2
For every H \in \operatorname {Ob} \mathcal {C}, there exists a local system H'_ \mathbb {Z} on S of free \mathbb {Z}-modules such that H'_ \mathbb {Z} \otimes \mathbb {Q} \approx H_ \mathbb {Q}.
1106Conditionhodge-theory-ii-4.2.2.4hodge-theory-ii-4.2.2.4.xml4.2.2.4hodge-theory-ii-4.2.2
For every H \in \operatorname {Ob} \mathcal {C}, the largest constant local system H^f of H is a constant family of Hodge sub-structures of H.
1112Lemmahodge-theory-ii-4.2.3hodge-theory-ii-4.2.3.xml4.2.3hodge-theory-ii-4.2
If \mathcal {C} satisfies and , then:
\mathcal {C} is a semi-simple abelian subcategory of the abelian subcategory of continuous families of Hodge \mathbb {Q}-structures on S.
If H \in \operatorname {Ob} \mathcal {C}, then its dual H^* and the \wedge ^p H are also in \mathcal {C};
if H_1,H_2 \in \operatorname {Ob} \mathcal {C}, then \operatorname {Hom} (H_1,H_2) \in \operatorname {Ob} \mathcal {C}.
1111Proofhodge-theory-ii-4.2.3
Let H \in \operatorname {Ob} \mathcal {C}, and let H_1 be a sub-object of H, in the category of continuous families of Hodge \mathbb {Q}-structures.
We will show that H_1 is a direct factor of H in this category.
We can suppose H to be homogeneous.
If \psi is a polarisation form for H, then the orthogonal of H_1 with respect to \psi is indeed a sub-object of H that is a supplement to H_1.
This proves (i)
If H \in \operatorname {Ob} \mathcal {C} is of weight n, then a polarisation of H defines an isomorphism between H^* and H \otimes \mathbb {Q} (n).
By , we thus have that H^* \in \operatorname {Ob} \mathcal {C}.
For arbitrary H \in \operatorname {Ob} \mathcal {C}, if we decompose H into its homogeneous components, then H^*= \bigoplus _n(H^n)^*, whence again H^* \in \operatorname {Ob} \mathcal {C}.
Finally, \wedge ^p H is a direct factor of \bigotimes ^p H, and \operatorname {Hom} (H_1,H_2) \simeq H_1^* \otimes H_2.
Let f \colon X \to S be a smooth proper morphism of reduced schemes.
The sheaf \mathrm {R} ^i f_* \mathbb {Q} is then a local system, and, for s \in S, ( \mathrm {R} ^i f_* \mathbb {Q} )_s \simeq \operatorname {H} ^i(X_s, \mathbb {Q} ) is endowed with a Hodge \mathbb {Q}-structure, which varies continuously in s.
1113Definitionhodge-theory-ii-4.2.4hodge-theory-ii-4.2.4.xml4.2.4hodge-theory-ii-4.2
Let S be a smooth connected scheme.
A continuous family H of Hodge \mathbb {Q}-structures on S^ \mathrm {an} is said to be algebraic if there exists a non-empty Zariski open U of S, an integer k, and a smooth projective morphism f \colon X \to U such that H|U is a direct factor of \mathrm {R} f_* \mathbb {Q} \otimes \mathbb {Q} (k).
1125Propositionhodge-theory-ii-4.2.5hodge-theory-ii-4.2.5.xml4.2.5hodge-theory-ii-4.2
The category of algebraic continuous families of Hodge \mathbb {Q}-structures on S satisfies the conditions of .
If a continuous family H of Hodge structures on S is such that its restriction to a dense Zariski open U of S is algebraic, then H is algebraic.
If f \colon X \to S is smooth and proper, then \mathrm {R} f_* \mathbb {Q} is algebraic.
1124Proofhodge-theory-ii-4.2.5
Claim (ii) is evident.
We will prove (i).
Let \mathcal {C}_0 be the set of continuous families of Hodge structures on S that are of the form \mathrm {R} f_* \mathbb {Q} (with f smooth and projective).
Then:
By the Künneth formula
\mathrm {R} (f \times g)_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Q} \otimes \mathrm {R} g_* \mathbb {Q} ,
\mathcal {C}_0 is stable under tensor products.
Since
\mathrm {R} (f \sqcup g)_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Q} \oplus \mathrm {R} g_* \mathbb {Q} ,
\mathcal {C}_0 is stable under direct sums.
The homogeneous components of each H \in \operatorname {Ob} \mathcal {C}_0 are polarisable (cf. ).
The objects H \in \operatorname {Ob} \mathcal {C}_0 satisfy , since
\mathrm {R} f_* \mathbb {Q} \simeq \mathrm {R} f_* \mathbb {Z} \otimes \mathbb {Q} .
The objects H \in \operatorname {Ob} \mathcal {C}_0 satisfy , by .
Let \mathcal {C}_1 be the set of direct factors of objects of \mathcal {C}_0.
Then \mathcal {C}_1 is stable under tensor products, direct sums, and direct factors, and satisfies , , and .
Furthermore, \mathbb {Q} (-1) is in \mathcal {C}_1, in the form of \mathrm {R} ^2 f_* \mathbb {Q} for f \colon \mathbb {P} _S^1 \to S the projective bundle.
The category \mathcal {C}_2 consisting of the H \otimes \mathbb {Q} (k) (for k \in \mathbb {N}) thus satisfy all the conditions of .
To thus deduce (i), it suffices to note that, if H is a continuous family of Hodge structures, and U a dense Zariski open of S, then a sub-object of, a polarisation on, or an integer lattice of H|U uniquely extend to H.
We now prove (iii).
If f \colon X \to S is smooth and projective, then there exists a dense Zariski open subset U of S along with a commutative diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X' \ar [rr,"p"] \ar [dr,swap,"f'"]
&& X|U \ar [dl,"f|U"]
\\ & U
\end {tikzcd}
with f' smooth and projective, and p birational and surjective (by applying Chow's lemma and the resolution of singularities to the generic fibre of f).
The restriction map
p^* \colon ( \mathrm {R} f_* \mathbb {Q} )|U \to \mathrm {R} f'_* \mathbb {Q}
is thus a direct injection of continuous families of Hodge structures, with left inverse given by the Gysin morphism p_!, whence the algebraicity of \mathrm {R} f_* \mathbb {Q}.
1126Theoremhodge-theory-ii-4.2.6hodge-theory-ii-4.2.6.xml4.2.6hodge-theory-ii-4.2
Let S be a connected topological space that is locally connected and locally simply connected, endowed with a basepoint s.
Let \mathcal {C} be a category of continuous Hodge structures on S that satisfy the conditions of .
Then, if H \in \operatorname {Ob} \mathcal {C}, the representation of \pi _1(S,s) on the fibre (H_ \mathbb {Q} )_s is semisimple.
1127Footnotehodge-theory-ii-4.2.6-footnote-1hodge-theory-ii-4.2.6-footnote-1.xml1hodge-theory-ii-4.2
Let S be a smooth connected scheme.
A family of Hodge structures on S is a continuous family H of Hodge structures on S that satisfies the following conditions:
The Hodge filtration on (H_ \mathbb {C} )_s varies holomorphically with s, i.e. it corresponds to a filtration F of H_ \mathcal {O} =H_ \mathbb {Z} \otimes \mathcal {O}.
The covariant derivative \nabla satisfies \nabla F^p \subset \Omega _S^1 \otimes F^{p-1}.
Let \mathcal {C} be the category of such continuous families of Hodge \mathbb {Q}-structures on S that underlie a family of Hodge structures and whose homogeneous direct factors are polarisable.
It is clear that \mathcal {C} satisfies , , and .
We can deduce from results of W. Schmid (August 1970, unpublished) and P.A. Griffiths [G1970] that \mathcal {C} satisfies .
This theorem, of which is a corollary, allows us to apply and its corollaries to the objects of \mathcal {C}.
Let H \in \operatorname {Ob} \mathcal {C}.
The local system H_ \mathbb {Q} defines a complex local system H_ \mathbb {C} =H_ \mathbb {Q} \otimes \mathbb {C}.
For every complex local system V on S, we denote by V^c the complex vector bundle that it defines, identified with the sheaf of its continuous sections.
By definition, the group S (from ) acts on H_ \mathbb {C} ^c.
A sub-bundle of H_ \mathbb {C} ^c is said to be horizontal, or locally constant, if it is defined by a local subsystem of H_ \mathbb {C}.
1129Lemmahodge-theory-ii-4.2.7hodge-theory-ii-4.2.7.xml4.2.7hodge-theory-ii-4.2
Let V be a rank-1 local subsystem of H_ \mathbb {C}.
Suppose that the n-th tensor power V^{ \otimes n} (for some n \geqslant1) of V is a trivial local system, i.e. that \pi _1(S,s) acts on V_s via a finite (necessarily cyclic) group.
Then, for all t \in S, tV^c is again locally constant.
1128Proofhodge-theory-ii-4.2.7
In order for tV^c to be locally constant, it suffices for (tV^c)^{ \otimes n}={tV^{c}}^{ \otimes n} \subset \bigotimes ^n H_ \mathbb {C} to be locally constant.
But V^{ \otimes n} is generated by a horizontal global section v, and, by hypothesis, tv is again horizontal .
We proceed by induction on \dim (H_ \mathbb {Q} )_s.
We can suppose H to be homogeneous and non-zero.
Let d be the minimal dimension of the non-zero complex local subsystems of H_ \mathbb {C}.
The sum W of all the local subsystems of H_ \mathbb {C} of dimension d (which are automatically simple) is "defined over \mathbb {Q}", i.e. is of the form W_ \mathbb {Q} \otimes \mathbb {C} for W_ \mathbb {Q} a local subsystem of H_ \mathbb {Q}.
By construction, W_s is a semisimple complex \pi _1(S,s)-module, so (W_ \mathbb {Q} )_s is a semisimple \pi _1(S,s)-module over \mathbb {Q}.
Let
1711hodge-theory-ii-1.3hodge-theory-ii-1.3.xmlHodge Theory II › Filtrations › The two filtrations lemma1.3hodge-theory-ii-1823hodge-theory-ii-1.3.1hodge-theory-ii-1.3.1.xml1.3.1hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with a filtration F.
The filtration is said to be biregular if it induces a finite filtration on each component of K.
We recall the definition of the terms E_r^{pq}(K,F), or simply E_r^{pq}, of the spectral sequence defined by F.
We set
Z_r^{pq} = \operatorname {Ker} (d \colon F^p(K^{p+q}) \to F^{p+q+1}/F^{p+r}(K^{p+q+1}))
and dually we define B_r^{pq} by the formula
K^{p+q}/B_r^{pq} = \operatorname {Coker} (d \colon F^{p-r+1}(K^{p+q+1}) \to K^{p+q}/F^{p+1}(K^{p+q})).
These formulas still make sense for r= \infty.
We note that the use here of the notation B_r^{pq} is different to that of Godement [G1958].
We have, by definition:
395Equationhodge-theory-ii-1.3.1.1hodge-theory-ii-1.3.1.1.xml1.3.1.1hodge-theory-ii-1.3.1 E_r^{pq} = \operatorname {Im} (Z_r^{pq} \to K^{p+q}/B_r^{pq}) \tag{1.3.1.1}
396Equationhodge-theory-ii-1.3.1.2hodge-theory-ii-1.3.1.2.xml1.3.1.2hodge-theory-ii-1.3.1 = Z_r^{pq}/(B_r^{pq} \cap Z_r^{pq}) \tag{1.3.1.2}
397Equationhodge-theory-ii-1.3.1.3hodge-theory-ii-1.3.1.3.xml1.3.1.3hodge-theory-ii-1.3.1 = \operatorname {Ker} (K^{p+q}/B_r^{pq} \to K^{p+q}/(Z_r^{pq}+B_r^{pq})). \tag{1.3.1.3}
We can thus write
398Equationhodge-theory-ii-1.3.1.4hodge-theory-ii-1.3.1.4.xml1.3.1.4hodge-theory-ii-1.3.1 \begin {aligned} B_r^{p \bullet } \cap Z_r^{p \bullet } & \coloneqq (dF^{p-r+1}+F^{p+1}) \cap (d^{-1}F^{p+r} \cap F^p) \\ &= (dF^{p-r+1} \cap F^p) + (F^{p+1} \cap d^{-1}F^{p+r}) \end {aligned} \tag{1.3.1.4}
since dF^{p-r+1} \subset d^{-1}F^{p+r} and F^{p+1} \subset F^p.
For r< \infty, the E_r form a complex graded by the degree p-r(p+q), and E_{r+1} can be expressed as the cohomology of this complex:
399Equationhodge-theory-ii-1.3.1.5hodge-theory-ii-1.3.1.5.xml1.3.1.5hodge-theory-ii-1.3.1 E_{r+1}^{pq} = \operatorname {H} (E_r^{p-r,q+r-1} \xrightarrow {d_r} E_r^{pq} \xrightarrow {d_r} E_r^{p+r,q-r+1}). \tag{1.3.1.5}
For r=0, we have
400Equationhodge-theory-ii-1.3.1.6hodge-theory-ii-1.3.1.6.xml1.3.1.6hodge-theory-ii-1.3.1 E_0^{ \bullet \bullet } = \operatorname {Gr} _F^ \bullet (K^ \bullet ). \tag{1.3.1.6} 828Propositionhodge-theory-ii-1.3.2hodge-theory-ii-1.3.2.xml1.3.2hodge-theory-ii-1.3
Let K be a complex endowed with a biregular filtration F.
The following conditions are equivalent:
The spectral sequence defined by F degenerates (E_1=E_ \infty).
The morphisms d \colon K^i \to K^{i+1} are strictly compatible with the filtrations.
827Proofhodge-theory-ii-1.3.2
We will prove this in the case where \mathscr {A} is a category of modules.
For fixed p and q, the hypothesis that the arrows d_r with domains E_r^{pq} be zero for r \geqslant1 implies that, if x \in F^p(K^{p+q}) satisfies dx \in F^{p+1}(K^{p+q+1}), then there exists y \in K^{p+q} such that dy=0 and such that x and y have the same image in E_1^{pq}.
Modifying y by a boundary, and setting z=x-y, we then have
\forall x \in F^p(K^{p+q}) \left [ dx \in F^{p+1}(K^{p+q+1}) \implies \exists z \text { s.t. } z \in F^{p+1}(K^{p+q}) \text { and } dz=dx \right ]
or, in other words,
F^{p+1}(K^{p+q+1}) \cap dF^p(K^{p+q}) = dF^{p+1}(K^{p+q}). \tag{1}
If this condition is satisfied for arbitrary p and q, then by induction on r we have
F^{p+r} \cap dF^p = dF^{p+r}
which, for large p+r, can be written as
F^p \cap dK = dF^p. \tag{2}
Claim (2) trivially implies (1), and is equivalent to (ii), which proves the proposition.
831hodge-theory-ii-1.3.3hodge-theory-ii-1.3.3.xml1.3.3hodge-theory-ii-1.3
If (K,F) is a filtered complex, we denote by \operatorname {Dec} (K) the complex K endowed with the shifted filtration
\operatorname {Dec} (F)^p K^n = Z_1^{p+n,-p}.
This filtration is compatible with the differentials:
\begin {aligned} dZ_1^{p+n,-p} & \subset F^{p+n+1}(K^{n+1}) \cap \operatorname {Ker} (d) \\ & \subset Z_ \infty ^{p+n+1,-p} \\ & \subset Z_1^{p+n+1,-p}. \end {aligned}
Since
829Equationhodge-theory-ii-1.3.3.1hodge-theory-ii-1.3.3.1.xml1.3.3.1hodge-theory-ii-1.3.3 \begin {aligned} Z_1^{p+1+n,-p-1} & \subset F^{p+1+n}(K^n) \\ & \subset B_1^{p+n,-p} \\ & \subset Z_1^{p+n,-p} \end {aligned} \tag{1.3.3.1}
the evident arrow from Z_1^{p+n,-p}/Z_1^{p+1+n,-p-1} to Z_1^{p+n,-p}/B_1^{p+n,-p} is a morphism
830Equationhodge-theory-ii-1.3.3.2hodge-theory-ii-1.3.3.2.xml1.3.3.2hodge-theory-ii-1.3.3 u \colon E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,-p}(K). \tag{1.3.3.2} 838Propositionhodge-theory-ii-1.3.4hodge-theory-ii-1.3.4.xml1.3.4hodge-theory-ii-1.3
The morphisms in form a morphism of graded complexes from E_0( \operatorname {Dec} (K)) to E_1(K).
This morphism induces an isomorphism on cohomology.
This morphism induces step-by-step (via ) isomorphisms of graded complexes E_r( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}(K) (for r \geqslant1).
837Proofhodge-theory-ii-1.3.4
Let F' be the filtration on K defined by
{F'}^p(K^n) = \operatorname {Dec} (F)^{p-n}(K^n) = Z_1^{p,n-p}.
We trivially have isomorphisms
836Equationhodge-theory-ii-1.3.4.1hodge-theory-ii-1.3.4.1.xml1.3.4.1hodge-theory-ii-1.3.4 E_r^{p,n-p}( \operatorname {Dec} (K)) = E_{r+1}^{p+n,-p}(K,F') \tag{1.3.4.1}
that are compatible with the d_r and with .
The map u comes from and from the identity map
(K,F') \to (K,F).
This proves (i), and it remains to show that, for r \geqslant2,
E_r^{pq}(K,F') \xrightarrow { \sim } E_r^{pq}(K,F).
We have
\begin {aligned} Z_r^{pq}(K,F') &= Z_r^{pq}(K,F) \qquad \text {for }r \geqslant1 \\ Z_r^{pq}(K,F') \cap B_r^{pq}(K,F') &= Z_r^{pq}(K,F) \cap B_r^{pq}(K,F) \qquad \text {for }r \geqslant2 \end {aligned}
and we can then apply .
839hodge-theory-ii-1.3.5hodge-theory-ii-1.3.5.xml1.3.5hodge-theory-ii-1.3
The construction in is not self-dual.
The dual construction consists of defining
\operatorname {Dec} ^ \bullet (F)^pK^n = B_1^{p+n-1,-p+1}.
We then have morphisms
E_0^{p,n-p}( \operatorname {Dec} (K)) \to E_1^{p+n,p}(K) \to E_0^{p,n-p}( \operatorname {Dec} ^ \bullet (K))
and, for r \geqslant1, isomorphisms
E_r^{p,n-p}( \operatorname {Dec} (K)) \xrightarrow { \sim } E_{r+1}^{p+n,p}(K) \xrightarrow { \sim } E_{r}^{p,n-p}( \operatorname {Dec} ^ \bullet (K)).
Recall that a morphism of complexes is said to be a quasi-isomorphism if it induces an isomorphism on cohomology.
843Definitionhodge-theory-ii-1.3.6hodge-theory-ii-1.3.6.xml1.3.6hodge-theory-ii-1.3
A morphism f \colon (K,F) \to (K',F') of filtered complexes with biregular filtrations is a filtered quasi-isomorphism if \operatorname {Gr} _F(f) is a quasi-isomorphism, i.e. if the E_1^{pq}(f) are isomorphisms.
A morphism f \colon (K,F,W) \to (K,F',W') of biregular bifiltered complexes is a bifiltered quasi-isomorphism if \operatorname {Gr} _F \operatorname {Gr} _W(f) is a quasi-isomorphism.
844hodge-theory-ii-1.3.7hodge-theory-ii-1.3.7.xml1.3.7hodge-theory-ii-1.3
Let K be a differential complex of objects of \mathscr {A}, endowed with two filtrations F and W.
Let E_r^{pq} be the spectral sequence defined by W.
The filtration F induces on E_r^{pq} various filtrations, which we will compare.
845hodge-theory-ii-1.3.8hodge-theory-ii-1.3.8.xml1.3.8hodge-theory-ii-1.3 identifies E_r^{pq} with a quotient of a sub-object of K^{p+q}.
The E_r^{pq} term is thusly given by endowing with a filtration F_d induced by F, called the first direct filtration.
846hodge-theory-ii-1.3.9hodge-theory-ii-1.3.9.xml1.3.9hodge-theory-ii-1.3
Dually, identifies E_r^{pq} with a sub-object of a quotient of K^{p+q}, whence a new filtration F_{d^*} induced by F, called the second direct filtration.
847Lemmahodge-theory-ii-1.3.10hodge-theory-ii-1.3.10.xml1.3.10hodge-theory-ii-1.3
On E_0 and E_1, we have F_d=F_{d^*}.
651Proofhodge-theory-ii-1.3.10
For r=0,1, we have B_r^{pq} \subset Z_r^{pq}, and we apply .
851hodge-theory-ii-1.3.11hodge-theory-ii-1.3.11.xml1.3.11hodge-theory-ii-1.3 identifies E_{r+1}^{pq} with a quotient of a sub-object of E_r^{pq}.
We define the recurrent filtration F_r on the E_r^{pq} by the conditions
On E_0^{pq}, F_r=F_d=F_{d^*}.
On E_{r+1}^{pq}, the recurrent filtration is that induced by the recurrent filtration of E_r^{pq}.
852hodge-theory-ii-1.3.12hodge-theory-ii-1.3.12.xml1.3.12hodge-theory-ii-1.3
Definitions and still make sense for r= \infty.
If the filtration on K is biregular, then the direct filtrations on E_ \infty ^{pq} coincide with those on E_r^{pq}=E_ \infty ^{pq} for large enough r, and we define the recurrent filtration on E_ \infty ^{pq} as agreeing with that on E_r^{pq} for large enough r.
The filtrations F and W each induce a filtration on H^ \bullet (K), and E_ \infty ^{ \bullet \bullet }= \operatorname {Gr} _W^ \bullet ( \operatorname {H} ^ \bullet (K)).
The filtration F on \operatorname {H} ^ \bullet (K) then induces on E_ \infty ^{pq} a new filtration.
859Propositionhodge-theory-ii-1.3.13hodge-theory-ii-1.3.13.xml1.3.13hodge-theory-ii-1.3
For the first direct filtration, the morphisms d_r are compatible with the filtrations.
If E_{r+1}^{pq} is considered as a quotient of a sub-object of E_r^{pq}, then the first direct filtration on E_{r+1}^{pq} is finer than the filtration F' induced by the first direct filtration on E_r^{pq}Y we have F_d(E_{r+1}^{pq}) \subset F'(E_{r+1}^{pq}).
Dually, the morphisms d_r are compatible with the second direct filtration, and the second direct filtration on E_{r+1}^{pq} is less fine than the filtration induced by that of E_r^{pq}.
F_d(E_r^{pq}) \subset F_r(E_r^{pq}) \subset F_{d^*}(E_r^{pq}).
On E_ \infty ^{pq}, the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K) is finer than the first direct filtration and less fine than the second.
858Proofhodge-theory-ii-1.3.13
Claim (i) is evident, (ii) is its dual, and (iii) follows by induction.
The first claim of (iv) is easy to verify, and the second is its dual.
860hodge-theory-ii-1.3.14hodge-theory-ii-1.3.14.xml1.3.14hodge-theory-ii-1.3
We denote by \operatorname {Dec} (K) (resp. \operatorname {Dec} ^ \bullet (K)) the complex K endowed with the filtrations \operatorname {Dec} (W) and F (resp. \operatorname {Dec} ^ \bullet (W) and F).
It is clear by that the isomorphism in sends the first direct filtration on E_r( \operatorname {Dec} (K)) to the second direct filtration on E_{r+1}(K) (for r \geqslant1).
The dual isomorphism sends the second direct filtration on E_r( \operatorname {Dec} ^ \bullet (K)) to the second direct filtration on E_{r+1}(K).
865Lemmahodge-theory-ii-1.3.15hodge-theory-ii-1.3.15.xml1.3.15hodge-theory-ii-1.3
If the filtration F is biregular, and if, on the \operatorname {Gr} _W^p(K), the morphisms d are strictly compatible with the filtration induced by F, then
The morphism of graded complexes filtered by F
u \colon \operatorname {Gr} _{ \operatorname {Dec} (W)}(K) \to E_1(K,W)
is a filtered quasi-isomorphism.
Dually, the morphism in
u \colon E_1(K,W) \to \operatorname {Gr} _{ \operatorname {Dec} ^ \bullet (W)}(K)
is a filtered quasi-isomorphism.
864Proofhodge-theory-ii-1.3.15
It suffices, by duality, to prove (i).
By and , the complex E_1(K,W) filtered by F is a quotient of the filtered complex \operatorname {Gr} _{ \operatorname {Dec} (W)}(K).
Let U be the filtered complex given by the kernel, which is acyclic by (ii) of .
The long exact sequence in cohomology associated to the exact sequence of complexes
0 \to \operatorname {Gr} _F(U) \to \operatorname {Gr} _F( \operatorname {Gr} _{ \operatorname {Dec} (W)}(K)) \to \operatorname {Gr} _F(E_1(K,W)) \to 0
shows that u is a filtered quasi-isomorphism if and only if \operatorname {Gr} _F(U) is an acyclic complex.
By , and since U is acyclic, this reduces to asking that the differentials of U be strictly compatible with the filtration F.
From we obtain that U is the sum over p of the complexes
(U^p)^n = B_1^{p+n,-p}/Z_1^{p+1+n,-p-1}
endowed with the filtration induced by F.
Each differential d of each of the complexes U^p fits into a commutative diagram of filtered objects of the following type, where, for simplicity, we have omitted the total or complementary degree:
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
B^p/Z^{p+1}
\ar [rrr,"d"]
\ar [dr]
&&& B^{p+1}/Z^{p+1}
\ar [dd]
\\ & \operatorname {Coim} (d)
\ar [r,"(5)"]
& \operatorname {Im} (d)
\ar [ur] \ar [dr]
\\ W^{p+1}/Z^{p+1}
\ar [uu]
\ar [ur]
\ar [rrr,near start,swap,"(3)"]
\ar [white,urr,near end,swap," \color {black}(4)"]
&&& B^{p+1}/W^{p+2}
\ar [d]
\\ W^{p+1}/W^{p+2}
\ar [u]
\ar [rrr,near start,swap,"(1)"]
\ar [white,urrr," \color {black}(2)"description]
&&& W^{p+1}/W^{p+2}
\end {tikzcd}
By hypothesis, the morphism (1) is strict.
Since the square (2) is exactly the canonical decomposition of (1), the arrow (3) is a filtered isomorphism.
The arrows of the trapezium (4) are isomorphisms;
they are thus filtered isomorphisms, since (3) is a filtered isomorphism.
The fact that (5) is a filtered isomorphism implies that d is strict.
This proves the lemma.
867Theoremhodge-theory-ii-1.3.16hodge-theory-ii-1.3.16.xml1.3.16hodge-theory-ii-1.3
Let K be a complex endowed with two filtrations, W and F, with the filtration F biregular.
Let r_0 \geqslant0 be an integer, and suppose that, for 0 \leqslant r<r_0, the differentials of the graded complex E_r(K,W) are strictly compatible with the filtration F.
Then, for r \leqslant r_0+1, F_d=F_r=F_{d^*} on E_r^{pq}.
866Proofhodge-theory-ii-1.3.16
We will prove the theorem by induction on r_0.
For r_0=0, the hypothesis is empty, and we apply and (iii) of .
For r_0 \geqslant1, by the inductive hypothesis, we have F_d=F_r=F_{d^*} on E_r^{pq} for r \leqslant r_0.
By , the morphism u \colon E_0( \operatorname {Dec} (K)) \to E_1(K) is a filtered quasi-isomorphism.
It thus induces a filtered isomorphism from \operatorname {H} ^ \bullet ( \operatorname {Dec} (K)) to \operatorname {H} ^ \bullet (E_1(K)):
u \colon (E_1( \operatorname {Dec} (K)),F_r) \xrightarrow { \sim } (E_2(K),F_r).
Step-by-step, we thus deduce that the canonical isomorphism from E_s( \operatorname {Dec} (K)) to E_{s+1} (for s \geqslant1) is a filtered isomorphism for the recurrent filtration.
On E_1( \operatorname {Dec} (K)), F_r=F_d (), and we already know () that u' is a filtered isomorphism
u' \colon (E_1( \operatorname {Dec} (K)),F_d) \xrightarrow { \sim } (E_2(K),F_d).
On E_2(K), we thus have F_d=F_r.
This, combined with the dual result, proves the theorem for r_0=1.
Suppose that r_0 \geqslant2.
Then the arrows d_1 of E_1(K) are strictly compatible with the filtrations, and thus so too are the arrows d_0 of E_0( \operatorname {Dec} (K)) (indeed, u induces an isomorphism of spectral sequences, and we apply the criterion from ).
For 0<s<r_0-1, the isomorphism (E_s( \operatorname {Dec} (K)),F_r) \cong (E_{s+1}(K),F_r) shows that the d_s are strictly compatible with the recurrent filtrations.
By the induction hypothesis, we thus have F_d=F_r on E_s( \operatorname {Dec} (K)) for s \leqslant s_0.
The isomorphism (E_s( \operatorname {Dec} (K)),F_d) \cong (E_{s+1}(K),F_d) () then shows that F_d=F_r on E_r(K) for r \leqslant r_0+1.
This, combined with the dual result, proves the theorem.
869Corollaryhodge-theory-ii-1.3.17hodge-theory-ii-1.3.17.xml1.3.17hodge-theory-ii-1.3
Under the general hypotheses of , suppose that, for all r, the differentials d_r are strictly compatible with the recurrent filtrations on the E_r.
Then, on E_ \infty, the filtrations F_d, F_r, and F_{d^*} agree, and coincide with the filtration induced by the filtration F of \operatorname {H} ^ \bullet (K).
868Proofhodge-theory-ii-1.3.17
This follows immediately from and (iv) of .