3306hodge-theory-iiihodge-theory-iii.xmlHodge Theory IIIVolumeIIIP. Deligne.
"Théorie de Hodge II".
Pub. Math. de l'IHÉS 44 (1974) pp. 5–77.
publications.ias.edu/node/3613101hodge-theory-iii-introductionhodge-theory-iii-introduction.xmlIntroductionhodge-theory-iiiThis article follows Hodge Theory I and Hodge Theory II;
the numbering of the sections here follows that of (which contains to ).In , we gave an introduction to the yoga that underlies and the present article .
However, and are logically independent of ;
they also do not contain all the results that were stated in .In , we introduced the Hodge theory of non-singular (not necessarily complete) algebraic varieties.
Here we treat the case of arbitrary singularities.
From [hodge-theory-iii-7 (?)] onwards, we will make essential use of results from to .In the case of a complete singular algebraic variety , the fundamental idea is to use the resolution of singularities to replace by a simplicial system of smooth projective schemes
(or, as we will say, by a smooth projective simplicial scheme).
The results of [hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] allow us to find such simplicial schemes that have, in a suitable sense, the same cohomology as .
We can then "express", via a spectral sequence, the cohomology of in terms of the cohomology of the .
Following the general principles of , we can then, on each , use classical Hodge theory, and obtain an induced mixed Hodge structure on the cohomology of .In the case of an arbitrary algebraic variety , we "replace" by a smooth simplicial scheme with compactification given by a smooth projective simplicial scheme .
We can further arrange things so that is a normal crossing divisor, given by a union of smooth divisors .
We thus "express", via a spectral sequence, the cohomology of in terms of the cohomology of the -fold intersections of the (for ), and we obtain an induced mixed Hodge structure on .[hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] constitute a summary (without proofs) of the theory of "cohomological descent".
This theory, in the more general framework of a topos, is laid out in [SD].In [hodge-theory-iii-7.1 (?)], we restate certain results obtained previously, but in the language of filtered derived categories;
[hodge-theory-iii-7.2 (?)] contains a simpler proof of the two-filtrations lemma than that of .The unappealing [hodge-theory-iii-8.1 (?)] is the heart of this work.
The essential result is [hodge-theory-iii-8.1.15 (?)].
In its proof, to be able to apply the two-filtrations lemma, we use the fact that every morphism of mixed Hodge structures is strictly compatible with the filtrations and ().
In [hodge-theory-iii-8.2 (?)], we define the mixed Hodge structure of (for a separated scheme).
We show that the Hodge numbers of can only be non-zero for .
For \dim X]]>, we have a more precise result ([hodge-theory-iii-8.2.4 (?)]).
If is smooth (resp. complete), they can only be non-zero if, further, (resp. ).
If is a complete subscheme of , with complement , with complete and smooth of dimension , then we can, by N. Katz, interpret this "duality" between the complete case and the smooth case as coming from the "Alexander duality"
From the functorial properties of the theory thus constructed, we obtain the useful corollaries [hodge-theory-iii-8.2.5 (?)] to [hodge-theory-iii-8.2.8 (?)], which we state in terms independent of any Hodge theory.In [hodge-theory-iii-8.3 (?)], we endow the cohomology of any simplicial scheme with a mixed Hodge structure.
This generality is not illusory.
We can interpret relative cohomology spaces as being the cohomology of suitable simplicial schemes.
Let be the classifying space of the underlying Lie group of an algebraic group .
We can interpret the cohomology of as being the cohomology of a suitable simplicial scheme.
In [hodge-theory-iii-9.1 (?)], after having calculated the mixed Hodge structure on the cohomology of (for linear), we thus obtain that of .
As a corollary, we find that, if a linear group acts on a non-singular complete variety , then the map
factors through .In [hodge-theory-iii-10 (?)], we interpret in terms of abelian schemes the mixed Hodge structures of pure degree (see [hodge-theory-iii-0.5 (?)]), and we consider in detail the of curves.The theory developed up until here is an absolute theory (we do not study the functors ), and deals only with constant coefficients.
I conjecture that, if is a variation of polarisable Hodge structures (in the sense of Griffiths [G1970a]) on a scheme , then the cohomology of with coefficients in the local system is endowed with a natural mixed Hodge structure.
I can only prove this when is complete.3102hodge-theory-iii-terminology-and-notationhodge-theory-iii-terminology-and-notation.xmlTerminology and notationhodge-theory-iii2104hodge-theory-iii-0.1hodge-theory-iii-0.1.xmlIII.0.1hodge-theory-iii-terminology-and-notationLet be a continuous map between topological spaces.
We say that is proper if it is proper in the sense of Bourbaki (i.e. universally closed) and furthermore separated (i.e. the diagonal of is closed).2105hodge-theory-iii-0.2hodge-theory-iii-0.2.xmlIII.0.2hodge-theory-iii-terminology-and-notationFollowing Gabriel and Zisman, we say simplicial where we would have previously said semi-simplicial.2106hodge-theory-iii-0.3hodge-theory-iii-0.3.xmlIII.0.3hodge-theory-iii-terminology-and-notationWe denote by a Noetherian subring of such that is a field.
The useful cases are , , or .2107hodge-theory-iii-0.4hodge-theory-iii-0.4.xmlIII.0.4hodge-theory-iii-terminology-and-notationA mixed Hodge -structure consists of an -module of finite type, a finite increasing filtration on the -vector space , and a finite decreasing filtration on the -vector space .
We demand that the be Hodge -structures.
For (resp. ), we recover (resp. );
the results of carry over as they are.2108hodge-theory-iii-0.5hodge-theory-iii-0.5.xmlIII.0.5hodge-theory-iii-terminology-and-notationLet be a subset of .
We say that a mixed Hodge structure is of degree if the Hodge numbers are zero for .2109hodge-theory-iii-0.6hodge-theory-iii-0.6.xmlIII.0.6hodge-theory-iii-terminology-and-notationFrom now on, we denote by what we previously denoted by ().2110hodge-theory-iii-0.7hodge-theory-iii-0.7.xmlIII.0.7hodge-theory-iii-terminology-and-notationUnless explicitly stated otherwise, scheme means "scheme of finite type over ", and a sheaf on a scheme is a sheaf on the underlying topological space of .3103hodge-theory-iii-5hodge-theory-iii-5.xmlCohomological descent5hodge-theory-iii3093hodge-theory-iii-5.1hodge-theory-iii-5.1.xmlSimplicial topological spaces5.1hodge-theory-iii-5This section starts with some reminders for which we can refer to the homotopical seminar of Strasbourg, 1963/64.2141hodge-theory-iii-5.1.1hodge-theory-iii-5.1.1.xml5.1.1hodge-theory-iii-5.1We will us the following notation, where are integers.
= the opposite category of a category
= the category of functors from to .
= the finite totally ordered set .
= the increasing injection such that (for ).
= the increasing surjection such that (for ).
= the unique map from to .
= the category whose objects are the (for ) and whose morphisms are the increasing maps between the .
= the full subcategory of whose objects are the (for ).
= the full subcategory of whose objects are the (for ).
= the full subcategory of whose objects are the (for ).For any category , we define a simplicial object (resp. -truncated simplicial object) of to be an object of (resp. of ).
Similarly, a cosimplicial object (resp. -truncated cosimplicial object) is an object of (resp. of ).For , the -skeleton functor is the restriction functor
and the -coskeleton functor
is the right adjoint to .
Let be a simplicial object of .
We also call the functor
the skeleton, and its right adjoint
is called the coskeleton with respect to .The coskeleton functors exist if finite projective limits exist in ;
the relative coskeleton functors exist if fibred products do;
we have
2142hodge-theory-iii-5.1.2hodge-theory-iii-5.1.2.xml5.1.2hodge-theory-iii-5.1If is a simplicial object of , then we set
.
2143hodge-theory-iii-5.1.3hodge-theory-iii-5.1.3.xml5.1.3hodge-theory-iii-5.1Let .
The constant simplicial object is the simplicial object for which and .
A simplicial (resp. -truncated simplicial) object of augmented over is a morphism (resp. ).
We identify such a simplicial object (resp. -truncated simplicial object) augmented over with the object of (resp. of ) such that .
We have that .
These objects will be denoted by notation of the form .
The relative coskeletons will be mostly used in this setting, and denoted or simply .2144hodge-theory-iii-5.1.4hodge-theory-iii-5.1.4.xml5.1.4hodge-theory-iii-5.1The coskeleton of the augmented -truncated simplicial object is the simplicial object augmented over whose components are the cartesian powers of in , i.e.
2145hodge-theory-iii-5.1.5hodge-theory-iii-5.1.5.xml5.1.5hodge-theory-iii-5.1Let be a continuous map between topological spaces, and a sheaf on and a sheaf on .
The set of -morphisms from to is the set .The data of a -morphism from to consists of the data of, for each pair of opens such that , a map ;
these maps must satisfy the condition
2146hodge-theory-iii-5.1.5.starhodge-theory-iii-5.1.5.star.xmlCondition*hodge-theory-iii-5.1.5For and such that and , the diagram
commutes.2147hodge-theory-iii-5.1.6hodge-theory-iii-5.1.6.xml5.1.6hodge-theory-iii-5.1A simplicial topological space is a simplicial object of the category whose objects are topological spaces and whose morphisms are continuous maps.A sheaf on a simplicial topological space consists of
a family of sheaves on the ;
for all , an -morphism from to .
We further require that .A morphism from to is a family of morphisms such that, for all , we have that .2151hodge-theory-iii-5.1.7hodge-theory-iii-5.1.7.xml5.1.7hodge-theory-iii-5.1For an open of , let .
For , , and such that , let be the map induced by .
We immediately see that the system of sets (indexed by and ) and of maps uniquely determines (cf. ).
For such a system to come from a sheaf on , it is necessary and sufficient that:
whenever this makes sense;
for any , the for form a sheaf on .
2155hodge-theory-iii-5.1.8hodge-theory-iii-5.1.8.xml5.1.8hodge-theory-iii-5.1This shows that sheaves on can be interpreted as sheaves on a suitable site, and, in particular, form a topos .
We freely use, for sheaves on , the modern terminology for sheaves on a site.
With this in mind, a sheaf of abelian groups (resp. of rings, ...) is a system of sheaves of abelian groups (resp. of rings, ...) on the along with the morphisms .2156hodge-theory-iii-5.1.9hodge-theory-iii-5.1.9.xmlExamples5.1.9hodge-theory-iii-5.1
Let be a simplicial analytic space.
The structure sheaves form a sheaf of rings on .
Let be a simplicial analytic space augmented over .
The sheaves form a sheaf of -modules on .
Its -th exterior power is the sheaf of -modules denoted .
The de Rham complexes form a complex of sheaves on .
Let be a sheaf of abelian groups on .
The canonical flasque Godement resolutions form a complex of sheaves on that is a resolution of .
Let be a topological space.
Sheaves on the constant simplicial space can be identified with cosimplicial sheaves on .
In particular, an abelian sheaf on defines a chain complex .
A complex of abelian sheaves on defines a double complex that we again denote by (here is the cosimplicial degree) and whose associated simple complex is
2162hodge-theory-iii-5.1.9.1hodge-theory-iii-5.1.9.1.xmlEquation5.1.9.1hodge-theory-iii-5.1.9
with differential given by
2163hodge-theory-iii-5.1.9.2hodge-theory-iii-5.1.9.2.xmlEquation5.1.9.2hodge-theory-iii-5.1.9
We denote by the second filtration of , i.e.
2164hodge-theory-iii-5.1.9.3hodge-theory-iii-5.1.9.3.xmlEquation5.1.9.3hodge-theory-iii-5.1.9
2165hodge-theory-iii-5.1.10hodge-theory-iii-5.1.10.xml5.1.10hodge-theory-iii-5.1Let be a morphism of simplicial topological spaces, with components .
If (resp. ) is a sheaf on (resp. on ), then (resp. ) is a sheaf on (resp. on );
we denote it by (resp. by ).
The functors and are adjoint to one another;
they are the inverse image and direct image morphisms of topos morphism .2166hodge-theory-iii-5.1.11hodge-theory-iii-5.1.11.xml5.1.11hodge-theory-iii-5.1Let be an augmented simplicial topological space.
If is a sheaf on , then "is" a sheaf on .
The functor has a right adjoint
2167hodge-theory-iii-5.1.11.1hodge-theory-iii-5.1.11.1.xmlEquation5.1.11.1hodge-theory-iii-5.1.11
The functors and are the inverse image and direct image morphisms of topos morphism .2168hodge-theory-iii-5.1.12hodge-theory-iii-5.1.12.xml5.1.12hodge-theory-iii-5.1Let be the constant simplicial space associated to , and the morphism defined by .
For an abelian sheaf on , we can identify with a cosimplicial sheaf on ((IV) of ).
For a complex of abelian sheaves on , if is the degree component of the restriction of to , then the components of the simple complex associated to the double complex defined by are
2169hodge-theory-iii-5.1.12.1hodge-theory-iii-5.1.12.1.xmlEquation5.1.12.1hodge-theory-iii-5.1.12
The spectral sequence defined by the filtration () of can be written as
2170hodge-theory-iii-5.1.12.2hodge-theory-iii-5.1.12.2.xmlEquation5.1.12.2hodge-theory-iii-5.1.122171hodge-theory-iii-5.1.13hodge-theory-iii-5.1.13.xml5.1.13hodge-theory-iii-5.1Let (the topological space consisting of a single point).
We see that, for a sheaf on , the are the components of a cosimplicial set .
The functor (of global sections) is the functor
2172hodge-theory-iii-5.1.13.1hodge-theory-iii-5.1.13.1.xmlEquation5.1.13.1hodge-theory-iii-5.1.13
If is a complex of abelian sheaves, we denote by the chain complex whose components are the cosimplicial abelian groups , and by the associated simple complex.3094hodge-theory-iii-5.2hodge-theory-iii-5.2.xmlCohomology of simplicial topological spaces5.2hodge-theory-iii-52244hodge-theory-iii-5.2.1hodge-theory-iii-5.2.1.xml5.2.1hodge-theory-iii-5.2To each simplicial topological space is associated a usual topological space , called its geometric realisation (see below).
In the particular case where the are discrete, we recover the usual notion of geometric realisation of a simplicial set.In [S1968], G. Segal defined the cohomology of with values in an abelian group as .
Under suitable hypotheses, the filtration of by successive skeletons gives a spectral sequence
424hodge-theory-iii-5.2.1.1hodge-theory-iii-5.2.1.1.xmlEquation5.2.1.1hodge-theory-iii-5.2.1One of the interests in using this definition is that it also applies, for example, to the definition of the groups .
We will adopt another definition, which is better adapted to sheaf-theoretic techniques.425#247unstable-247.xmlGeometric realisationhodge-theory-iii-5.2.1For an integer , we denote by the simple in whose vertices are the set of basis vectors.
We identify with the set of vertices of .
Every function extends by linearity to .Let be a simplicial topological space.
Let
Let be the finest equivalence relation on for which, for any in , any , and any , we have
The geometric realisation of is by definition
2245hodge-theory-iii-5.2.2hodge-theory-iii-5.2.2.xmlDefinition5.2.2hodge-theory-iii-5.2Let be a simplicial topological space.
The functors of cohomology with coefficients in the abelian sheaf on are the derived functors of the functor ().This definition is equivalent to the following, sometimes more convenient.2246hodge-theory-iii-5.2.3hodge-theory-iii-5.2.3.xml5.2.3hodge-theory-iii-5.2Let be an abelian sheaf on .
We can show (by example, using (III) of ) that always admits resolutions on the right such that
0. ]]>
If is such a resolution, then we canonically have that
2189hodge-theory-iii-5.2.3.1hodge-theory-iii-5.2.3.1.xmlEquation5.2.3.1hodge-theory-iii-5.2.3
It is easy to show directly that the right-hand side is independent (up to unique isomorphism) of the choice of ;
for what follows, we are free to define by .
We can further show that the spectral sequence of the complex , filtered by ( and for )
2190hodge-theory-iii-5.2.3.2hodge-theory-iii-5.2.3.2.xmlEquation5.2.3.2hodge-theory-iii-5.2.3
is independent (up to unique isomorphism) of the choice of (cf. ).2247hodge-theory-iii-5.2.4hodge-theory-iii-5.2.4.xml5.2.4hodge-theory-iii-5.2We will need to make precise the above construction when passing to derived categories, and to give a relative variant of it.Let , let be the constant simplicial space defined by with augmentation map , and let be the map induced by .
Then
2248hodge-theory-iii-5.2.4.1hodge-theory-iii-5.2.4.1.xmlEquation5.2.4.1hodge-theory-iii-5.2.4The derived version of this equation is
2249hodge-theory-iii-5.2.4.2hodge-theory-iii-5.2.4.2.xmlEquation5.2.4.2hodge-theory-iii-5.2.4
where is the bounded-below derived category of the category of abelian sheaves on .
We will calculate and .2250hodge-theory-iii-5.2.5hodge-theory-iii-5.2.5.xml5.2.5hodge-theory-iii-5.2Let be a morphism of simplicial topological spaces.
The functor can be calculated "component by component": if is a complex of abelian sheaves on , then to calculate we take a resolution such that the components of satisfy for 0]]>, (cf. (III) of );
then .2251hodge-theory-iii-5.2.6hodge-theory-iii-5.2.6.xml5.2.6hodge-theory-iii-5.2The functor sends acyclic complexes to acyclic complexes.
It thus trivially derives to
If is an injective sheaf on , then the chain complex is a resolution of .
We thus obtain an isomorphism , and becomes
2252hodge-theory-iii-5.2.6.1hodge-theory-iii-5.2.6.1.xmlEquation5.2.6.1hodge-theory-iii-5.2.62253hodge-theory-iii-5.2.7hodge-theory-iii-5.2.7.xml5.2.7hodge-theory-iii-5.2Combined with and specialised to the case where , this equation proves .
In concrete terms, this implies that, to calculate , we can proceed in two steps:
We take a resolution such that the components of satisfy for 0]]>.
The complex (the derived category of the category of cosimplicial abelian sheaves on ) can be identified with .
is the simple complex associated to the double complex .
The spectral sequence generalises to a spectral sequence
2195hodge-theory-iii-5.2.7.1hodge-theory-iii-5.2.7.1.xmlEquation5.2.7.1hodge-theory-iii-5.2.7
induced by .3095hodge-theory-iii-5.3hodge-theory-iii-5.3.xmlCohomological descent5.3hodge-theory-iii-52310hodge-theory-iii-5.3.1hodge-theory-iii-5.3.1.xml5.3.1hodge-theory-iii-5.3Let be an augmented simplicial topological space.
For every sheaf on , we have a morphism
from the adjunction .
This morphism derives to a morphism of functors from to , namely
2308hodge-theory-iii-5.3.1.1hodge-theory-iii-5.3.1.1.xmlEquation5.3.1.1hodge-theory-iii-5.3.12311hodge-theory-iii-5.3.2hodge-theory-iii-5.3.2.xmlDefinition5.3.2hodge-theory-iii-5.3We say that is of cohomological descent if, for every abelian sheaf on , we have
and
0. ]]>This is equivalent to asking that be an isomorphism.2312hodge-theory-iii-5.3.3hodge-theory-iii-5.3.3.xml5.3.3hodge-theory-iii-5.3If is of cohomological descent, then, for , the canonical map
2276hodge-theory-iii-5.3.3.1hodge-theory-iii-5.3.3.1.xmlEquation5.3.3.1hodge-theory-iii-5.3.3
is an isomorphism.
In particular, for an abelian sheaf on , we have a spectral sequence ()
2277hodge-theory-iii-5.3.3.2hodge-theory-iii-5.3.3.2.xmlEquation5.3.3.2hodge-theory-iii-5.3.3For a complex, we again have, in hypercohomology, a spectral sequence
2278hodge-theory-iii-5.3.3.3hodge-theory-iii-5.3.3.3.xmlEquation5.3.3.3hodge-theory-iii-5.3.3In both cases, the (for fixed ) form a simplicial group, and
2313hodge-theory-iii-5.3.4hodge-theory-iii-5.3.4.xmlDefinition5.3.4hodge-theory-iii-5.3A continuous map is of cohomological descent if the augmentation morphism of , namely
is of cohomological descent.
We say that is of universal cohomological descent if, for every , the continuous map is of cohomological descent.2314hodge-theory-iii-5.3.5hodge-theory-iii-5.3.5.xml5.3.5hodge-theory-iii-5.3The fundamental results, proven in [SD], are the following.
The continuous maps of universal cohomological descent form a Grothendieck topology on the category of topological spaces, which we call the universal cohomological descent topology.
A proper () surjective map is of universal cohomological descent.
A map that admits sections locally on is of universal cohomological descent.
Let be a -truncated augmented simplicial space (with ).
For , let be the evident map.
We say that is a -truncated hypercover of , for the universal cohomological descent topology, if the maps
2117hodge-theory-iii-5.3.5.1hodge-theory-iii-5.3.5.1.xmlEquation5.3.5.1hodge-theory-iii-5.3.5
(for ) are of universal cohomological descent.
If is such a hypercover, then the simplicial space augmented over is of cohomological descent.
Let be a morphism of simplicial topological spaces augmented over .
a>> Y_\bullet \\@VxVV @VVyV \\S @= S \end {CD} ]]>
We say that is a hypercover for the universal cohomological descent topology if the evident maps are of universal cohomological descent.
If is such a hypercover then, for every ,
2315hodge-theory-iii-5.3.6hodge-theory-iii-5.3.6.xml5.3.6hodge-theory-iii-5.3For , (IV) of implies that the are of cohomological descent if the are of universal cohomological descent.
For , these maps are
For , is the subspace of consisting of the triples such that , , and .
The map is .For , (IV) of is .2316hodge-theory-iii-5.3.7hodge-theory-iii-5.3.7.xmlExample5.3.7hodge-theory-iii-5.3Let be an open cover, or a finite locally closed cover, of .
Let .
Then is of cohomological descent.
The spectral sequence in for is then exactly the Leray spectral sequence of the cover .2317hodge-theory-iii-5.3.8hodge-theory-iii-5.3.8.xml5.3.8hodge-theory-iii-5.3Let be as in (IV) of .
We say that is a proper -truncated hypercover of if the arrows in are proper and surjective.
For , we simply say "proper hypercover".3104hodge-theory-iii-6hodge-theory-iii-6.xmlExamples of simplicial topological spaces6hodge-theory-iii3087hodge-theory-iii-6.1hodge-theory-iii-6.1.xmlClassifying spaces6.1hodge-theory-iii-62379hodge-theory-iii-6.1.1hodge-theory-iii-6.1.1.xml6.1.1hodge-theory-iii-6.1Let be a continuous map.
For every sheaf on , the sheaf is endowed with a "descent data" with respect to , i.e. we have an isomorphism between the two inverse images of on , and this isomorphism satisfies a cocycle condition.
If admits a section locally on , then this construction defines an equivalence between the category of sheaves on and that of sheaves on endowed with a descent data.Take to be a (left) principal homogeneous space for the group on a space .
Then the -equivariant sheaves on are exactly the sheaves endowed with a descent data: every equivariant sheaf on is, in a unique way (as an equivariant sheaf), the inverse image of a sheaf on .2380hodge-theory-iii-6.1.2hodge-theory-iii-6.1.2.xml6.1.2hodge-theory-iii-6.1If a topological group acts on a space , then acts on by
We denote by the simplicial space
2344hodge-theory-iii-6.1.2.1hodge-theory-iii-6.1.2.1.xmlEquation6.1.2.1hodge-theory-iii-6.1.2
If is a principal homogeneous space for the group on , then the map
identifies with the iterated fibre product :
In particular, we have (by (III) of )
2347hodge-theory-iii-6.1.2.2hodge-theory-iii-6.1.2.2.xmlEquation6.1.2.2hodge-theory-iii-6.1.2
(for a principal homogeneous space).
For all , is a principal homogeneous space for the group on .
For every equivariant sheaf on , is an equivariant sheaf on ;
by , the latter is the inverse image of on .
Every equivariant sheaf on thus defines a sheaf on .
It is easy to show that we thus obtain an equivalence between the category of equivariant sheaves on and the category of sheaves on that satisfy
2349hodge-theory-iii-6.1.starhodge-theory-iii-6.1.star.xmlProperty*hodge-theory-iii-6.1.2 For every , the structure morphism is an isomorphism.
Construction (b) above is natural in .
We set
2351hodge-theory-iii-6.1.2.3hodge-theory-iii-6.1.2.3.xmlEquation6.1.2.3hodge-theory-iii-6.1.2
(mixed cohomology of with coefficients in ).
Under the hypotheses of (a), if is the inverse image of on , then
2352hodge-theory-iii-6.1.2.4hodge-theory-iii-6.1.2.4.xmlEquation6.1.2.4hodge-theory-iii-6.1.2
(for a principal homogeneous space).
This generalises (which is the case ).
2381hodge-theory-iii-6.1.3hodge-theory-iii-6.1.3.xml6.1.3hodge-theory-iii-6.1Let be the topological space consisting of a single point.
We define the simplicial classifying space of , denoted , to be the simplicial space
Let be a principal homogeneous space for the group on .
The evident morphism
defines a composite morphism
2382hodge-theory-iii-6.1.3.1hodge-theory-iii-6.1.3.1.xmlEquation6.1.3.1hodge-theory-iii-6.1.3We will see below that, in good cases, , and that the image of consists of the characteristic classes of .2383hodge-theory-iii-6.1.4hodge-theory-iii-6.1.4.xml6.1.4hodge-theory-iii-6.1Let be a Lie group, a classifying space for , and the universal principal homogeneous -space.
Let be a -space;
then is a principal homogeneous -space over such that, for every equivariant sheaf on , is the inverse image of a sheaf on .
Since is contractible, and by , we have
2384hodge-theory-iii-6.1.4.1hodge-theory-iii-6.1.4.1.xmlEquation6.1.4.1hodge-theory-iii-6.1.4In particular, for ,
2385hodge-theory-iii-6.1.4.2hodge-theory-iii-6.1.4.2.xmlEquation6.1.4.2hodge-theory-iii-6.1.4We can see that the isomorphism in is a particular case of where and .2386hodge-theory-iii-6.1.5hodge-theory-iii-6.1.5.xml6.1.5hodge-theory-iii-6.1The spectral sequence in for
is essentially the Eilenberg–Moore spectral sequence.
We briefly recall how it allows us to relate the rational cohomologies of and of , for connected .
The algebra is a connected graded Hopf algebra of finite dimension over .
If is the graded module of its primitive elements, then we have
and the generators of are of odd degree.
The simplicial algebra is
which is the exterior algebra of the suspension of the constant cosimplicial module ;
we thus have (by Quillen [Q1968]) that .
The pages are only zero for even;
we thus have that , and, for a suitable filtration, we canonically have that
and non-canonically that
2387hodge-theory-iii-6.1.6hodge-theory-iii-6.1.6.xml6.1.6hodge-theory-iii-6.1Let be a (complex) linear algebraic group.
If is a maximal torus of , with Weyl group , then
325hodge-theory-iii-6.1.6.1hodge-theory-iii-6.1.6.1.xmlEquation6.1.6.1hodge-theory-iii-6.1.6
If is a torus with character group , then
327hodge-theory-iii-6.1.6.2hodge-theory-iii-6.1.6.2.xmlEquation6.1.6.2hodge-theory-iii-6.1.6
(an isomorphism of graded Hopf algebras).
We will only use (a) in the following weaker form:
(The splitting principle).
The map is injective.
For completion, we recall a proof of (a').If is a Borel subgroup of , then the bundle on is a fibre in flag spaces.
By [2.1 and 2.6.3, D1968], which is better explained in [Proposition 3.1, G1970], or by [B1956], the Leray spectral sequence of degenerates to rational cohomology.
We thus have that .
We conclude by noting that .3088hodge-theory-iii-6.2hodge-theory-iii-6.2.xmlConstruction of hypercovers6.2hodge-theory-iii-62442hodge-theory-iii-6.2.1hodge-theory-iii-6.2.1.xml6.2.1hodge-theory-iii-6.2Let be a simplicial set.
Denote by the set of increasing surjective maps from to (degeneracy operators), and set
2439hodge-theory-iii-6.2.1.1hodge-theory-iii-6.2.1.1.xmlEquation6.2.1.1hodge-theory-iii-6.2.1Recall that, for all , the map
2440hodge-theory-iii-6.2.1.2hodge-theory-iii-6.2.1.2.xmlEquation6.2.1.2hodge-theory-iii-6.2.1
is bijective.2443hodge-theory-iii-6.2.2hodge-theory-iii-6.2.2.xmlDefinition6.2.2hodge-theory-iii-6.2We say that a simplicial topological space is -split if the maps in are homeomorphisms.Let be a -truncated simplicial set.
For , we again define by , and then is a bijection.
We say that a -truncated simplicial topological space is -split if is a homeomorphism for .2444hodge-theory-iii-6.2.3hodge-theory-iii-6.2.3.xml6.2.3hodge-theory-iii-6.2For an -split -truncated topological simplicial space augmented over , let be the triple consisting of , , and the evident map from to .This triple satisfies the following:
2445hodge-theory-iii-6.2.starhodge-theory-iii-6.2.star.xmlProperty*hodge-theory-iii-6.2.3 is an -split -truncated simplicial topological space augmented over , and is a continuous map from to .2446hodge-theory-iii-6.2.4hodge-theory-iii-6.2.4.xmlProposition6.2.4hodge-theory-iii-6.2Let satisfy above.
Up to unique isomorphism, there exists exactly one -split -truncated topological space augmented over such that .
It is equivalent to give either or:
a morphism ; and
a morphism , such that the diagram
\beta >> (\operatorname {cosq} Y)_{n+1} \\@V{f''}VV @VV{f'}V \\Z_{n+1} @>>> (\operatorname {cosq}\operatorname {sq}_n Z)_{n+1} \end {CD} ]]>
commutes.
2453#245unstable-245.xmlProofhodge-theory-iii-6.2.4
[5.1.3, SD].
This proposition also applies to simplicial objects in other categories apart from that of topological spaces; it suffices that satisfy the following:
2454hodge-theory-iii-6.2.4.1hodge-theory-iii-6.2.4.1.xml6.2.4.1hodge-theory-iii-6.2.4Finite projective limits exist in .
Finite projective sums exist in , and they are disjoint and universal.2455hodge-theory-iii-6.2.5hodge-theory-iii-6.2.5.xml6.2.5hodge-theory-iii-6.2 allows us to construct, by induction, proper hypercovers of .
We take , proper and surjective.
Then is a -truncated proper hypercover of (), and it is -split.
We take , i.e. .
Applying , we associate to the -split -truncated augmented simplicial topological space
Suppose to be chosen such that
is proper and surjective (for example, if is proper and surjective).
Then is an -split -truncated proper hypercover of .
Assume that we have already constructed an -split -truncated proper hypercover .
We take and, applying , we associate to an -split -truncated augmented semi-simplicial space .
Suppose that is such that
is proper and surjective (for example, if is proper and surjective).
Then is an -split -truncated proper hypercover of .
The thus constructed are the successive skeletons of an -split proper hypercover of .
2456hodge-theory-iii-6.2.6hodge-theory-iii-6.2.6.xml6.2.6hodge-theory-iii-6.2We say that a simplicial scheme over is smooth if the are smooth;
it is said to be proper if the are compact.
A normal crossing divisor of , assumed to be smooth, is a family of normal crossing divisors () such that the form a simplicial subscheme of .
This definition is justified by the following lemma.2457hodge-theory-iii-6.2.7hodge-theory-iii-6.2.7.xmlLemma6.2.7hodge-theory-iii-6.2If is a normal crossing divisor of , then the logarithmic de Rham complexes , endowed with the weight filtration (), form a filtered complex on .
1565#246unstable-246.xmlProofhodge-theory-iii-6.2.7
This follows from (ii) of .
We denote the complex by .2458hodge-theory-iii-6.2.8hodge-theory-iii-6.2.8.xml6.2.8hodge-theory-iii-6.2Using , we can show that, for every separated scheme over , there exists:
a simplicial scheme over , smooth and proper, that we can take to be -split;
a normal crossing divisor of ; we set ;
an augmentation that realises as a proper hypercover of .
Furthermore, any two such systems are covered by a third, and a morphism can be covered by a morphism
of such systems (see [SD]).3089hodge-theory-iii-6.3hodge-theory-iii-6.3.xmlRelative cohomology6.3hodge-theory-iii-62511hodge-theory-iii-6.3.1hodge-theory-iii-6.3.1.xml6.3.1hodge-theory-iii-6.3The mapping cone construction for morphisms of simplicial sets works in the same way for simplicial objects in any category that has a final object and finite sums.
For , the cone satisfies
We take to be:
the category of topological spaces and continuous maps, with final object ;
the category of pairs where is a topological space and is an abelian sheaf on , with an arrow consisting of a continuous map and a -morphism () , with final object .
2512hodge-theory-iii-6.3.2hodge-theory-iii-6.3.2.xml6.3.2hodge-theory-iii-6.3Let be a morphism of topological simplicial spaces, with cone .
Let be an abelian sheaf on and an abelian sheaf on , and let be a -morphism.
The cone of is an abelian sheaf on , and we set
2513hodge-theory-iii-6.3.2.1hodge-theory-iii-6.3.2.1.xmlEquation6.3.2.1hodge-theory-iii-6.3.2
These are the relative cohomology groups.
We can easily show that they fit into a long exact sequence
2514hodge-theory-iii-6.3.2.2hodge-theory-iii-6.3.2.2.xmlEquation6.3.2.2hodge-theory-iii-6.3.22515hodge-theory-iii-6.3.3hodge-theory-iii-6.3.3.xml6.3.3hodge-theory-iii-6.3More generally, let and be bounded-below complexes of abelian sheaves on and (respectively), and let be a -morphism.
We thus obtain a complex on .
We again define the hypercohomology
These groups appear in an exact sequence analogous to that of , coming, in the suitable derived category, from a distinguished triangle
2516hodge-theory-iii-6.3.3.1hodge-theory-iii-6.3.3.1.xmlEquation6.3.3.1hodge-theory-iii-6.3.3
2517hodge-theory-iii-6.3.4hodge-theory-iii-6.3.4.xml6.3.4hodge-theory-iii-6.3The construction presented above is not the only one possible.
It has the inconvenience that, even if we start with true topological spaces and (i.e. constant simplicial spaces), we are led to consider non-constant simplicial spaces.3090hodge-theory-iii-6.4hodge-theory-iii-6.4.xmlMultisimplicial spaces6.4hodge-theory-iii-62535hodge-theory-iii-6.4.1hodge-theory-iii-6.4.1.xml6.4.1hodge-theory-iii-6.4Let be an integer.
An -simplicial object in a category is a contravariant functor from the -fold product to .
The diagonal simplicial object is the composite functor .2536hodge-theory-iii-6.4.2hodge-theory-iii-6.4.2.xml6.4.2hodge-theory-iii-6.4As in , we define the topos of sheaves on an -simplicial topological space.
Let be the functor
from sheaves on to -cosimplicial sets.
For small , we often prefer to write (with -many copies of ).
We have a co-augmentation .
The functors of cohomology with values in an abelian sheaf are the derived functors of the "global sections" functor .
They can be calculated by a procedure parallel to that of , i.e.
2219hodge-theory-iii-6.4.2.1hodge-theory-iii-6.4.2.1.xmlEquation6.4.2.1hodge-theory-iii-6.4.2A sheaf on an -simplicial topological space induces a sheaf on the diagonal simplicial space .
It follows from the Cartier–Eilenberg–Zilber theorem that
2220hodge-theory-iii-6.4.2.2hodge-theory-iii-6.4.2.2.xmlEquation6.4.2.2hodge-theory-iii-6.4.22537hodge-theory-iii-6.4.3hodge-theory-iii-6.4.3.xml6.4.3hodge-theory-iii-6.4We restrict to the case .
A bisimplicial object -augmented over a simplicial object is a contravariant functor from to such that is the composite functor
To denote a bisimplicial object -augmented over , with underlying bisimplicial object , we will use notation of the form
For , is a simplicial object augmented over .
If is a sheaf on , then the form a sheaf on ;
we thus define a topos morphism
We explain in [SD] that can be calculated "component by component", i.e.
It thus follows that if, for each , the morphism is of cohomological descent, then is of cohomological descent: for every complex of abelian sheaves, we have
2538hodge-theory-iii-6.4.4hodge-theory-iii-6.4.4.xml6.4.4hodge-theory-iii-6.4In [SD], we show that, for every separated simplicial scheme , there exists a bisimplicial scheme that is -augmented over , along with such that:
The are smooth and projective;
is the complement of a normal crossing divisor in , and we can suppose the to be a union of smooth divisors.
For , is a proper hypercover of , and we can take it to be -split.
The construction proceeds as in , but the induction is more complicated.
The claims of uniqueness () remain true, mutatis mutandis.3105hodge-theory-iii-7hodge-theory-iii-7.xmlSupplements to §17hodge-theory-iii3106hodge-theory-iii-7.1hodge-theory-iii-7.1.xmlFiltered derived category7.1hodge-theory-iii-7This section completes .2559hodge-theory-iii-7.1.1hodge-theory-iii-7.1.1.xml7.1.1hodge-theory-iii-7.1Let be an abelian category.
We set:
(resp. )
= the category of filtered (resp. bifiltered) objects with finite filtration(s) of .
(resp. )
= the category of filtered (resp. bifiltered) bounded-below complexes of objects of , up to homotopy that respects the filtration(s).
(resp. )
= the triangulated category induced from (resp. from ) by inverting the filtered (resp. bifiltered) quasi-isomorphisms ();
this is the derived filtered category (resp. derived bifiltered category).2560hodge-theory-iii-7.1.2hodge-theory-iii-7.1.2.xml7.1.2hodge-theory-iii-7.1A filtered quasi-isomorphism induces an isomorphism of spectral sequences .
An object of thus defines a spectral sequence .
Similarly, an object of defines an accumulation of spectral sequences of the type considered in .2561hodge-theory-iii-7.1.3hodge-theory-iii-7.1.3.xml7.1.3hodge-theory-iii-7.1Let be a left exact functor from to an abelian category .
Suppose that every object of injects into an injective object.
The functor can then be "derived" to give the functors
1166hodge-theory-iii-7.1.3.1hodge-theory-iii-7.1.3.1.xmlEquation7.1.3.1hodge-theory-iii-7.1.3
1167hodge-theory-iii-7.1.3.2hodge-theory-iii-7.1.3.2.xmlEquation7.1.3.2hodge-theory-iii-7.1.3
1168hodge-theory-iii-7.1.3.3hodge-theory-iii-7.1.3.3.xmlEquation7.1.3.3hodge-theory-iii-7.1.3
They can be calculated as follows: if is a -acyclic resolution (resp. a filtered resolution, resp. a bifiltered resolution) of ( and ), then .The hypercohomology spectral sequence (for ) of is the spectral sequence of (cf. ).2562hodge-theory-iii-7.1.4hodge-theory-iii-7.1.4.xml7.1.4hodge-theory-iii-7.1We will need more precise results for the functors , where is an augmentation of a simplicial topological space.
The case , where will suffice.We reuse the notation of , and recall :
For every complex , the simple complex is endowed with a natural filtration ().
A quasi-isomorphism induces a filtered quasi-isomorphism .
Then factors as
and factors as
The spectral sequence of the filtered complex is exactly .2563hodge-theory-iii-7.1.5hodge-theory-iii-7.1.5.xml7.1.5hodge-theory-iii-7.1If is filtered (resp. bifiltered), then is filtered (resp. bifiltered): we have
2564hodge-theory-iii-7.1.5.1hodge-theory-iii-7.1.5.1.xmlEquation7.1.5.1hodge-theory-iii-7.1.5
2565hodge-theory-iii-7.1.5.2hodge-theory-iii-7.1.5.2.xmlEquation7.1.5.2hodge-theory-iii-7.1.52566hodge-theory-iii-7.1.6hodge-theory-iii-7.1.6.xml7.1.6hodge-theory-iii-7.1Let be a complex of cosimplicial sheaves on , endowed with an increasing filtration .
We define the diagonal filtration of and to be the increasing filtration of given by
2567hodge-theory-iii-7.1.6.1hodge-theory-iii-7.1.6.1.xmlEquation7.1.6.1hodge-theory-iii-7.1.6We have
2568hodge-theory-iii-7.1.6.2hodge-theory-iii-7.1.6.2.xmlEquation7.1.6.2hodge-theory-iii-7.1.6The functor sends filtered quasi-isomorphisms to filtered quasi-isomorphisms, and defines
2569hodge-theory-iii-7.1.6.3hodge-theory-iii-7.1.6.3.xmlEquation7.1.6.3hodge-theory-iii-7.1.6
whence, by composition with ,
2570hodge-theory-iii-7.1.6.4hodge-theory-iii-7.1.6.4.xmlEquation7.1.6.4hodge-theory-iii-7.1.6
From , we see that
2571hodge-theory-iii-7.1.6.5hodge-theory-iii-7.1.6.5.xmlEquation7.1.6.5hodge-theory-iii-7.1.62572hodge-theory-iii-7.1.7hodge-theory-iii-7.1.7.xml7.1.7hodge-theory-iii-7.1If is a bifiltered complex of cosimplicial sheaves, then is endowed with the three filtrations , , and , and defines different bifiltered complexes.For example, for increasing , the functor sends bifiltered quasi-isomorphisms to bifiltered quasi-isomorphisms, and thus defines
2573hodge-theory-iii-7.1.7.1hodge-theory-iii-7.1.7.1.xmlEquation7.1.7.1hodge-theory-iii-7.1.7
By composition with , we thus obtain
2574hodge-theory-iii-7.1.7.2hodge-theory-iii-7.1.7.2.xmlEquation7.1.7.2hodge-theory-iii-7.1.7
and we have that
2575hodge-theory-iii-7.1.7.3hodge-theory-iii-7.1.7.3.xmlEquation7.1.7.3hodge-theory-iii-7.1.7
in .3107hodge-theory-iii-7.2hodge-theory-iii-7.2.xmlSupplements to the two filtrations lemma7.2hodge-theory-iii-7In this section, we give a new proof of the two filtrations lemma () and some supplements.1043hodge-theory-iii-7.2.1hodge-theory-iii-7.2.1.xml7.2.1hodge-theory-iii-7.2Let be a bounded-below bifiltered complex with objects in an abelian category , with assumed to be biregular.We say that is -splitable if the filtered complex can be written as a sum of filtered complexes
with
Let be an integer or .
The following condition was considered in and :1044hodge-theory-iii-7.2.2hodge-theory-iii-7.2.2.xmlCondition7.2.2hodge-theory-iii-7.2For every non-negative integer , the differentials of the graded complex are strictly compatible with the recurrent filtration defined by .1045hodge-theory-iii-7.2.3hodge-theory-iii-7.2.3.xml7.2.3hodge-theory-iii-7.2It is clear that, if is -splitable, then is satisfied for .
Conversely, it seems that if is satisfied for , then everything is as if the functor were exact.
For example, we will show that the of the spectral sequence can then be identified with the spectral sequence , and that the spectral sequence degenerates ().1046hodge-theory-iii-7.2.4hodge-theory-iii-7.2.4.xml7.2.4hodge-theory-iii-7.2We immediately deduce from the definition () that the first direct filtration of is the filtration of by the images
Dually, the second direct filtration () of is the filtration of by the kernels
The recurrent filtration of is intermediary between these two filtrations ((iii) of ).1047hodge-theory-iii-7.2.5hodge-theory-iii-7.2.5.xmlProposition7.2.5hodge-theory-iii-7.2Suppose that satisfies for some .
Then
also satisfies for .
For , the sequence
is exact;
for , the sequence
is exact.
In particular, for , the two direct filtrations and the recurrent filtration of agree.
1051#243unstable-243.xmlProofhodge-theory-iii-7.2.5
Fix an integer .
We will prove the claim
1052#242unstable-242.xmlhodge-theory-iii-7.2.5 If , then injects into ;
if , then its image is .
by induction on .
The claim is always true.
Assume , and let us prove .
We can suppose that .
We have a diagram
and the image of the vertical inclusions is .
Since is injective,
If , then is strictly compatible with , whence
and thus injects into .
This proves .
The statements , combined with the dual statements, implies (ii).
It follows from (ii) that, for and , injects into , and that the differential of is strictly compatible with the first direct filtration of .
We thus deduce, by induction on , that the first direct filtration of coincides with the recurrent filtration, and (i) then follows.
1053hodge-theory-iii-7.2.6hodge-theory-iii-7.2.6.xml7.2.6hodge-theory-iii-7.2If is satisfied for , and , then we denote by the filtration of .
The exact sequences from (ii) of
define, for , an (autodual) isomorphism that is compatible with the differentials , namely
1054hodge-theory-iii-7.2.6.1hodge-theory-iii-7.2.6.1.xmlEquation7.2.6.1hodge-theory-iii-7.2.61055hodge-theory-iii-7.2.7hodge-theory-iii-7.2.7.xml7.2.7hodge-theory-iii-7.2If is satisfied for , then the sequences
are exact for all .
If the filtration is biregular, then the sequence
is thus exact.
We can rewrite this sequence as
1056hodge-theory-iii-7.2.7.1hodge-theory-iii-7.2.7.1.xmlLemma7.2.7.1hodge-theory-iii-7.2.7Let be a complex endowed with a biregular filtration .
For to be acyclic, it is necessary and sufficient that be acyclic and that the differentials of be strictly compatible with the filtration .
1057#244unstable-244.xmlProofhodge-theory-iii-7.2.7.1
This is a particular case of .
Applying , taking to be the complex
we see that:
This sequence is exact, i.e. the differentials of are strictly compatible with the filtration ;
the filtration of induced by the filtration of coincides with the filtration induced by the filtration of ;
an analogous statement holds for .
In conclusion:1062hodge-theory-iii-7.2.8hodge-theory-iii-7.2.8.xmlProposition7.2.8hodge-theory-iii-7.2If satisfies for , then the spectral sequence degenerates ().
Furthermore, we have an isomorphism of spectral sequences
3108hodge-theory-iii-8hodge-theory-iii-8.xmlHodge theory of algebraic spaces8hodge-theory-iii3109hodge-theory-iii-8.1hodge-theory-iii-8.1.xmlHodge complexes8.1hodge-theory-iii-81362hodge-theory-iii-8.1.1hodge-theory-iii-8.1.1.xml8.1.1hodge-theory-iii-8.1Let be as in .
Then a Hodge -complex of weight consists of
a complex such that is an -module of finite type for all ;
a filtration on , i.e. a filtered complex , and an isomorphism in .
The following axioms have to be satisfied:
1366hodge-theory-iii-8.1.1-ch1hodge-theory-iii-8.1.1-ch1.xmlAxiomCH1hodge-theory-iii-8.1.1The differential of is strictly compatible with the filtration ;
in other words, the spectral sequence defined by degenerates at (i.e. ) .
1367hodge-theory-iii-8.1.1-ch2hodge-theory-iii-8.1.1-ch2.xmlAxiomCH2hodge-theory-iii-8.1.1For all , the filtration on defines a Hodge -structure of weight on , i.e. the filtration is -opposite to its complex conjugate (which makes sense, since ).1368hodge-theory-iii-8.1.2hodge-theory-iii-8.1.2.xml8.1.2hodge-theory-iii-8.1Let be a topological space.
Then a cohomological Hodge -complex of weight on consists of:
a complex ,
a filtered complex ,
an isomorphism in .
The following axiom has to be satisfied:
1373hodge-theory-iii-8.1.2-chchodge-theory-iii-8.1.2-chc.xmlAxiomCHChodge-theory-iii-8.1.2The triple is a Hodge complex of weight .When , we simply speak of Hodge complexes and cohomological Hodge complexes.The following statement reformulates a part of classical Hodge theory (cf. ).1374hodge-theory-iii-8.1.3hodge-theory-iii-8.1.3.xmlScholium8.1.3hodge-theory-iii-8.1Let be a complex Kähler variety.
Let be the complex consisting of the constant sheaf in degree , and let be the holomorphic de Rham complex , with its stupid filtration.
We have (the Poincaré lemma), and is a cohomological Hodge complex of weight .1375hodge-theory-iii-8.1.4hodge-theory-iii-8.1.4.xmlRemark8.1.4hodge-theory-iii-8.1If is a Hodge complex (resp. cohomological Hodge complex) of weight , then is a Hodge complex (resp. cohomological Hodge complex) of weight .1376hodge-theory-iii-8.1.5hodge-theory-iii-8.1.5.xml8.1.5hodge-theory-iii-8.1A mixed Hodge -complex consists of:
a complex such that is an -module of finite type for all ;
a filtered complex (for an increasing filtration ) and an isomorphism in ;
a bifiltered complex (for an increasing filtration and a decreasing filtration ) and an isomorphism in .
The following axiom must be satisfied:
1381hodge-theory-iii-8.1.5-chmhodge-theory-iii-8.1.5-chm.xmlAxiomCHMhodge-theory-iii-8.1.5For all , the system consisting of the complex , the complex filtered by , and the isomorphism
is a Hodge -complex of weight .1382hodge-theory-iii-8.1.6hodge-theory-iii-8.1.6.xml8.1.6hodge-theory-iii-8.1A cohomological mixed Hodge -complex on a topological space consists of:
a complex such that is an -module of finite type for all , and such that ;
a filtered complex (for an increasing filtration ) and an isomorphism in ;
a bifiltered complex (for an increasing filtration and a decreasing filtration ) and an isomorphism in .
The following axiom must be satisfied:
1387hodge-theory-iii-8.1.6-chmchodge-theory-iii-8.1.6-chmc.xmlAxiomCHMChodge-theory-iii-8.1.6For all , the system consisting of the complex , the complex filtered by , and the isomorphism
is a cohomological Hodge -complex of weight .We can trivially show:1388hodge-theory-iii-8.1.7hodge-theory-iii-8.1.7.xmlProposition8.1.7hodge-theory-iii-8.1If is a cohomological mixed Hodge -complex, then is a mixed Hodge -complex.1389hodge-theory-iii-8.1.8hodge-theory-iii-8.1.8.xmlScholium8.1.8hodge-theory-iii-8.1Let be the complement in a smooth proper scheme , of a normal crossing divisor , with the inclusion morphism.
Let be the canonical filtration on .
We have ()
Let be the weight filtration of , and its Hodge filtration (by the ).
Then gives an isomorphism in , and, by and ,
is a cohomological mixed Hodge complex on .The following result is an "abstract" version of .1390hodge-theory-iii-8.1.9hodge-theory-iii-8.1.9.xmlScholium8.1.9hodge-theory-iii-8.1Let be a mixed Hodge -complex.
On the pages of the spectral sequence of , the recurrent filtration and the two direct filtrations defined by all coincide.
The filtration () on and the filtration on define a mixed Hodge -structure on ( along with ).
The morphisms are compatible with the Hodge bigrading;
in particular, they are strictly compatible with the Hodge filtration.
The spectral sequence of degenerates at (i.e. ).
The spectral sequence of degenerates at (i.e. ).
The spectral sequence of the complex , endowed with the filtration induced by , degenerates at .
717#240unstable-240.xmlProofhodge-theory-iii-8.1.9
The proof of (i), (ii), (iii), (iv) is parallel to that of .
The analogy of and has been taken as an axiom ( and ).
The proofs of , , and can be carried over as they are, and, as in , we thus deduce (i), (ii), (iii), and (iv).
Claims (v) and (vi) then follow from .
1391hodge-theory-iii-8.1.10hodge-theory-iii-8.1.10.xml8.1.10hodge-theory-iii-8.1We abbreviate "differential graded" to DG, and "bounded-below differential graded" to DG+.
A DG complex of -modules can be seen as a double complex of -modules;
the first degree is that of the complex, and the second degree is that defined by the grading of DG modules.
We denote by the derived category of the category of bounded-below complexes of DG -modules, with degrees uniformly bounded below.A DG mixed Hodge -complex consists of:
a complex ;
a filtered complex , and is isomorphism in ;
a bifiltered complex , and an isomorphism in .
We require that, for each , the component of whose second degree is be a mixed Hodge -complex.We denote by the filtration by the second degree of .
We denote by the filtration of induced by shifting () .
We will not confuse the filtration that this induces on (also denoted by ) with the shifted filtration of the filtration induced on by .1093hodge-theory-iii-8.1.10.1hodge-theory-iii-8.1.10.1.xmlVariation8.1.10.1hodge-theory-iii-8.1.10We similarly define cosimplicial (resp. -cosimplicial) mixed Hodge -complexes by replacing "DG" with "cosimplicial" (resp. -cosimplicial) everywhere in the definition.
The functor from cosimplicial -modules to DG -modules sends cosimplicial mixed Hodge -complexes to DG mixed Hodge -complexes (and similarly for -cosimplicial).1392hodge-theory-iii-8.1.11hodge-theory-iii-8.1.11.xml8.1.11hodge-theory-iii-8.1Let be a simplicial topological space.
A cohomological mixed Hodge -complex on consists of:
a complex ;
a filtered complex , and an isomorphism in ;
a bifiltered complex , and an isomorphism in .
The following axiom has to be satisfied:
1397hodge-theory-iii-8.1.11-chmbullethodge-theory-iii-8.1.11-chmbullet.xmlAxiomCHM●hodge-theory-iii-8.1.11The restriction of to each of the is a cohomological mixed Hodge -complex.For , we simply speak of a cohomological mixed Hodge complex.1398hodge-theory-iii-8.1.12hodge-theory-iii-8.1.12.xmlExample8.1.12hodge-theory-iii-8.1Let be a smooth proper scheme (over ), and let be a normal crossing divisor in .
We set , and we denote by the inclusion of into .
The constructions in can be performed "uniformly in ", as and show.
They thus give a cohomological mixed Hodge complex
on .1399hodge-theory-iii-8.1.13hodge-theory-iii-8.1.13.xml8.1.13hodge-theory-iii-8.1Let be a cohomological mixed Hodge -complex on .
Write to mean the category of cosimplicial -modules.
If we apply the functor to , then we obtain
a cosimplicial complex ;
a cosimplicial filtered complex ;
a cosimplicial bifiltered complex ;
isomorphisms as in (b) and (c) of between these objects.
It is clear that the described above is a cosimplicial mixed Hodge -module.
It defines a DG mixed Hodge -complex ().1405hodge-theory-iii-8.1.14hodge-theory-iii-8.1.14.xml8.1.14hodge-theory-iii-8.1Let be a DG mixed Hodge -complex, and consider the spectral sequence abutting to the cohomology of , which is defined by the filtration .
By hypothesis, the pages of this spectral sequence are endowed with mixed Hodge -structures.
The content of is that every spectral sequence, including its abutment, is endowed with a mixed Hodge -structure.
In other words, is a spectral sequence in the abelian category of mixed Hodge -structures.
The structure on the abutment corresponds to a natural mixed Hodge -complex "structure" on .1406hodge-theory-iii-8.1.15hodge-theory-iii-8.1.15.xmlTheorem8.1.15hodge-theory-iii-8.1Let be a DG mixed Hodge -complex.
With the notation of , , and , is a mixed Hodge -complex.
We have
1410hodge-theory-iii-8.1.15.1hodge-theory-iii-8.1.15.1.xmlEquation8.1.15.1hodge-theory-iii-8.1.15
where the complex is the simple complex associated to the double complex of Hodge -structures of weight given by
1411hodge-theory-iii-8.1.15.2hodge-theory-iii-8.1.15.2.xmlEquation8.1.15.2hodge-theory-iii-8.1.15{\partial }>> \operatorname {H}^{b-\beta }(\operatorname {Gr}_\beta ^W K^{\bullet ,\gamma +1}) @>{\partial }>> \operatorname {H}^{b-(\beta -1)}(\operatorname {Gr}_{\beta -1}^W K^{\bullet ,\gamma +1}) \\@. @A{d''}AA @AA{d''}A \\@. \operatorname {H}^{b-\beta }(\operatorname {Gr}_\beta ^W K^{\bullet ,\gamma }) @>>{\partial }> \operatorname {H}^{b-(\beta -1)}(\operatorname {Gr}_{\beta -1}^W K^{\bullet ,\gamma }) \end {CD} \tag{8.1.15.2} ]]>
On the pages of the spectral sequence defined by , the recurrent filtration and the two direct filtrations defined by agree;
similarly, on the , the recurrent filtration and the two direct filtrations induced by agree.
We denoted these filtrations again by and (respectively).
For , is a mixed Hodge -structure ().
The differentials are morphisms of mixed Hodge -structures.
The filtration on is a filtration in the category of mixed Hodge structures, and
1415hodge-theory-iii-8.1.15.3hodge-theory-iii-8.1.15.3.xmlEquation8.1.15.3hodge-theory-iii-8.1.15
1416hodge-theory-iii-8.1.16hodge-theory-iii-8.1.16.xmlRemark8.1.16hodge-theory-iii-8.1In , the filtration () plays a more important role than .
We have
1417hodge-theory-iii-8.1.16.1hodge-theory-iii-8.1.16.1.xmlEquation8.1.16.1hodge-theory-iii-8.1.16
and can be rewritten as
1418hodge-theory-iii-8.1.16.2hodge-theory-iii-8.1.16.2.xmlEquation8.1.16.2hodge-theory-iii-8.1.16
In the same spirit, we note that, with the notation of ,
depends only on , and not on the choice of compactification .1419hodge-theory-iii-8.1.17hodge-theory-iii-8.1.17.xml8.1.17hodge-theory-iii-8.1We now prove .
Claims (i) and (ii) follow from the fact that
(this isomorphism is compatible with , and we apply ).
The proof of is left to the reader.Consider the claims:
The morphisms of are strictly compatible with the recurrent filtration defined by .
The morphisms of are strictly compatible with the recurrent filtration defined by .
, endowed with the filtrations considered in (a) and (b) is a mixed Hodge -structure.
The are morphisms of mixed Hodge -structures.
We will prove (a) and (b) for , and (c) and (d) for , by simultaneous induction.
Proof of (a).
We have
By (iv) of , the spectral sequence of degenerates at .
By , the spectral sequence of degenerates at , and this proves (a) ().
Proof of (b).
We have
We apply (v) of and .
Proof of (c).
We have to show that , endowed with the filtrations induced by on and by on , is a mixed Hodge -structure.
But induces on the same filtration as , and so we conclude by (ii) of .
Proof of (c)+(d)(a)+(b)+(c).
This is the key point of the proof, and is a simple application of (along with ).
Proof of (a)+(b)+(c)(d).
By , the recurrent filtration and the direct filtrations induced by (resp. ) on agree for , and we apply (i) of .
This finishes the proof by induction of (a), (b), (c), and (d).Claim (iii) was proven in part (E) of the proof by induction, and the claims of (iv) are (c) and (d).
We will prove in the form of .The fact that
follows from (a) and .
The fact that
follows from (b) and .We conclude by the following lemma, whose proof is left to the reader:1420hodge-theory-iii-8.1.18hodge-theory-iii-8.1.18.xmlLemma8.1.18hodge-theory-iii-8.1Let be a mixed Hodge -structure.
For a filtration of to come from a filtration of , in the abelian category of mixed Hodge -structures, it is necessary and sufficient that, for all , be a mixed Hodge -structure.1421hodge-theory-iii-8.1.19hodge-theory-iii-8.1.19.xml8.1.19hodge-theory-iii-8.1We now apply and to .We obtain a mixed Hodge complex consisting of (a), (b), and (c) below.
The complex , whose cohomology groups are the .
The filtration of .
With the notation of , we have
The page of the corresponding spectral sequence is the sum of the groups of form ;
this contributes to the ;
it corresponds to for :
1425hodge-theory-iii-8.1.19.1hodge-theory-iii-8.1.19.1.xmlEquation8.1.19.1hodge-theory-iii-8.1.19
The bifiltration of .
The filtration endows with a Hodge structure of weight , and the differentials of are the morphisms of the Hodge -structures.
The complex is the simple complex associated the double complex of Hodge structures of weight given by:
{\mathrm {Gysin}}>> \operatorname {H}^{b-2r}(\widetilde {Y}_{q+1}^{r},\varepsilon _\mathbb {Q}^{r})[-r] @>{\mathrm {Gysin}}>> \operatorname {H}^{b-2(r-2)}(\widetilde {Y}_{q+1}^{r-1},\varepsilon _\mathbb {Q}^{r-1})[-r+1] \\ @. @A{\sum _i(-1)^i\delta _i}AA @AA{\sum _i(-1)^i\delta _i}A \\ @. \operatorname {H}^{b-2r}(\widetilde {Y}_q^r,\varepsilon _\mathbb {Q}^r)[-r] @>>{\mathrm {Gysin}}> \operatorname {H}^{b-2(r-2)}(\widetilde {Y}_q^{r-1},\varepsilon _\mathbb {Q}^{r-1})[-r+1] \end {CD} ]]>
The indices in the terms in satisfy .In particular, the are endowed with mixed Hodge structures ().1427hodge-theory-iii-8.1.20hodge-theory-iii-8.1.20.xmlProposition8.1.20hodge-theory-iii-8.1Under the above hypotheses:
The mixed Hodge structure of is functorial in the pair .
In rational cohomology, the spectral sequence degenerates at (i.e. ) and abuts to the weight filtration of .
The Hodge structure of induced by that of is also that of .
The Hodge numbers of satisfy
For , the Hodge numbers of satisfy
1433#241unstable-241.xmlProofhodge-theory-iii-8.1.20
Both (i) and (ii) follow from the general theory ().
To prove (iii), we contemplate and note that the Hodge structures (for ) satisfy (iii).
To prove (iv), we note that the Hodge structures (for ) satisfy (iv).
1434hodge-theory-iii-8.1.21hodge-theory-iii-8.1.21.xml8.1.21hodge-theory-iii-8.1We define a cohomological mixed Hodge -complex on an -simplicial topological space by transposing .
If is such a complex, then we see, as in , that is an -cosimplicial mixed Hodge -complex ().Let be a smooth and proper -simplicial scheme.
A normal crossing divisor on is a system of normal crossing divisors on the such that is an -simplicial subscheme of .
Let .
We define, as in , a cohomological mixed Hodge complex on .We can, as in , thus deduce a mixed Hodge structure on , but, since
this does not teach us anything new (the above equality would be an identity between mixed Hodge structures).1435hodge-theory-iii-8.1.22hodge-theory-iii-8.1.22.xml8.1.22hodge-theory-iii-8.1Set .
For every bounded-below bicosimplicial complex (where is the differential degree, and is the -th simplicial degree), we denote by the cosimplicial complex of simplicial degree given by
We denote by the filtration on that, on each , induces the filtration by the simplicial degree .Let be a bicosimplicial mixed Hodge -complex.
It follows from that is a cosimplicial mixed Hodge complex.
Let be the filtration of by the simplicial degree .
By and , the spectral sequence defined by comes from a spectral sequence in the abelian category of mixed Hodge -structures
1436hodge-theory-iii-8.1.22.1hodge-theory-iii-8.1.22.1.xmlEquation8.1.22.1hodge-theory-iii-8.1.22In this spectral sequence, the mixed Hodge -structure on is the mixed Hodge -structure ((ii) of ) of the cohomology of the mixed Hodge -complex .
The mixed Hodge -structure of the abutment is the mixed Hodge -structure of the cohomology of the mixed Hodge -complex .
The filtration is given by
1437hodge-theory-iii-8.1.23hodge-theory-iii-8.1.23.xml8.1.23hodge-theory-iii-8.1In particular, a cohomological mixed Hodge -complex on a bisimplicial topological space defines a spectral sequence of mixed Hodge -structures
1438hodge-theory-iii-8.1.23.1hodge-theory-iii-8.1.23.1.xmlEquation8.1.23.1hodge-theory-iii-8.1.23In the particular case considered in , for , we have , and can be written as
1439hodge-theory-iii-8.1.23.2hodge-theory-iii-8.1.23.2.xmlEquation8.1.23.2hodge-theory-iii-8.1.231440hodge-theory-iii-8.1.24hodge-theory-iii-8.1.24.xml8.1.24hodge-theory-iii-8.1If and are mixed Hodge -complexes, then then their product , defined by
is again a mixed Hodge -complex.Let (resp. ) be the complement of a normal crossing divisor (resp. ) in a smooth and proper scheme (resp. ).
Then is the complement of a normal crossing divisor in .
Let (resp. , ) be the inclusion of (resp. , ) into (resp. , ).We have a quasi-isomorphism
a filtered quasi-isomorphism
and a bifiltered morphism
Applying the functor gives us, by the Künneth formula, and by its analogue for cohomology of coherent analytic sheaves, an isomorphism of mixed Hodge complexes
whence an isomorphism of mixed Hodge -structures
1441hodge-theory-iii-8.1.25hodge-theory-iii-8.1.25.xml8.1.25hodge-theory-iii-8.1Standard arguments, based on the Cartier–Eilenberg–Zilber theorem, give us the following simplicial variant of .Let (resp. ) be the complement of a normal crossing divisor in a smooth and proper simplicial scheme (resp. ).
Let and .We have an isomorphism of mixed Hodge complexes
and, in particular, an isomorphism of mixed Hodge -structures
1442hodge-theory-iii-8.1.26hodge-theory-iii-8.1.26.xml8.1.26hodge-theory-iii-8.1Let (resp. ) be the complement of a normal crossing divisor in a smooth and proper bisimplicial scheme (resp. ).
Let and .
Standard arguments show that, in rational cohomology, the spectral sequence for is the tensor product of the spectral sequences for and .3110hodge-theory-iii-8.2hodge-theory-iii-8.2.xmlSeparated algebraic spaces8.2hodge-theory-iii-82858hodge-theory-iii-8.2.1hodge-theory-iii-8.2.1.xml8.2.1hodge-theory-iii-8.2Let be a scheme (or an algebraic space) of finite type over , and assume it to be separated.
Then there exists a diagram
2354hodge-theory-iii-8.2.1.1hodge-theory-iii-8.2.1.1.xmlEquation8.2.1.1hodge-theory-iii-8.2.1
in which is the complement of a normal crossing divisor in a smooth and proper simplicial scheme , and is a proper hypercover of .
We have ((II) of )
2355hodge-theory-iii-8.2.1.2hodge-theory-iii-8.2.1.2.xmlEquation8.2.1.2hodge-theory-iii-8.2.1In , we endow with a mixed Hodge structure.
Let be the mixed Hodge structure on induced by .2859hodge-theory-iii-8.2.2hodge-theory-iii-8.2.2.xmlProposition8.2.2hodge-theory-iii-8.2With the above notation, the mixed Hodge structure on is independent of the choice of .
We call it the mixed Hodge structure of .
For every morphism of schemes, is a morphism of mixed Hodge structures.
1653#238unstable-238.xmlProofhodge-theory-iii-8.2.2
The proof uses , and is parallel to that of (C) of (cf. also [hodge-theory-iii-8.3.3 (?)]).
2860hodge-theory-iii-8.2.3hodge-theory-iii-8.2.3.xml8.2.3hodge-theory-iii-8.2For smooth , we can show, by taking to be a constant simplicial scheme, that the mixed Hodge structure in coincides with that defined in .2861hodge-theory-iii-8.2.4hodge-theory-iii-8.2.4.xmlTheorem8.2.4hodge-theory-iii-8.2The pairs such that the Hodge number of are non-zero satisfy the following conditions:
.
If , and , then
If is proper, then .
If is smooth, then .
The same conclusion holds if is an equidimensional "rational homology manifold" of dimension , i.e. if, for all ,
] (0,0) to (3,0) node[label={below:$p$}]{};
\draw [->] (0,0) to (0,3) node[label={left:$q$}]{};
\node [below] at (2,0) {\scriptsize $N$};
\node [left] at (0,2) {\scriptsize $N$};
\node [below] at (1.3,0) {\scriptsize $n$};
\node [left] at (0,1.3) {\scriptsize $n$};
\draw [dashed] (2,0) to (2,2.7);
\draw [dashed] (0,2) to (2.7,2);
\draw (0,1.3) to (1.3,1.3) to (1.3,0) to cycle;
\node at (0.5,0.2) {\scriptsize proper};
\node at (0.8,1.1) {\scriptsize smooth};
\end {scope}
\begin {scope}[shift={(4,0)}]
\draw [->] (0,0) to (4,0) node[label={below:$p$}]{};
\draw [->] (0,0) to (0,3) node[label={left:$q$}]{};
\node [below] at (2.35,0) {\scriptsize $N\qquad n$};
\node [left] at (0,2) {\scriptsize $N$};
\node [left] at (0,2.7) {\scriptsize $n$};
\draw [dashed] (2,0) to (2,2.7);
\draw [dashed] (0,2) to (2.7,2);
\draw [dashed] (2.7,0) to (0,2.7);
\draw (0.7,2) to (2,2) to (2,0.7) to (0.7,0.7) to cycle;
\draw (0.7,2) to (2,0.7);
\node at (1.2,0.9) {\scriptsize proper};
\node at (1.5,1.8) {\scriptsize smooth};
\end {scope}
\end {tikzpicture}]]>
1694#239unstable-239.xmlProofhodge-theory-iii-8.2.4
Claim (ii) will be proven in [hodge-theory-iii-8.3.10 (?)].
Claim (i) follows from (iii) of .
For proper , we can find a diagram as in in which , and (iii) then follows from (iv) of .
For smooth , (iv) follows from and .
The general case then follows: if is a resolution of singularities of , then is injective, since the Poincaré dual transpose of is a retraction for it.
2862hodge-theory-iii-8.2.5hodge-theory-iii-8.2.5.xmlProposition8.2.5hodge-theory-iii-8.2SupposeReferences3307G1970G1970.xmlPeriods of integrals on algebraic manifolds, IIIReference1970P.A. GriffithsPubl. Math. IHÉS 38 pp. 125–1803309G1970aG1970a.xmlPeriods of integrals on algebraic manifolds. Summary of main results and discussion of open problemsReference1970P.A. GriffithsBull. Am. Math. Soc. 76 pp. 288–2963311S1968S1968.xmlClassifying spaces and spectral sequencesReference1968G. SegalPubl. Math. IHÉS 34 pp. 105–1123313Q1968Q1968.xmlNotes on the homology of commutative ringsReference1968D.G. QuillenM.I.T.3315D1968D1968.xmlThéorème de Lefschetz et critères de dégénérescence de suites spectralesReference1968P. DelignePubl. Math. IHÉS 35 pp.107–1263317B1956B1956.xmlSur les variétés analytiques complexesReference1956A. BlanchardAnn. Sc. E.N.S. 733319SDSD.xmlTechniques de descente cohomologiqueReferenceB. Saint-DonatSGA 4.V bisBacklinks3321indexindex.xmlDeligne's "Hodge Theory I, II, and III"TO-DO: replace fref by refThis 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]The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
You can view the entire source code of this translation (and contribute or submit corrections) in the GitHub repository.
Corrections and comments welcome.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 of abelian schemes
Hodge Theory III
Introduction ✓
Terminology and notation ✓
Cohomological descent ✓
Simplicial topological spaces
Cohomology of simplicial topological spaces
Cohomological descent
Examples of simplicial topological spaces ✓
Classifying spaces
Construction of hypercovers
Relative cohomology
Multi-simplicial spaces
Supplements to §1 ✓
Filtered derived category
Supplements to the two-filtrations lemma
Hodge theory of algebraic spaces
Hodge complexes ✓
Separated algebraic spaces
Hodge theory of simplicial schemes
Examples and applications
Cohomology of groups and of classifying spaces
Hodge theory of smooth hypersurfaces, following Griffiths
Construction of complexes of first-order differential operators
Hodge theory in level
1-motives
1-motives and bi-extensions
Algebraic interpretation of the mixed : the case of curves
Translation of a theorem of Picard
Bibliography3322hodge-theory-iii-introductionhodge-theory-iii-introduction.xmlHodge Theory III › Introductionhodge-theory-iiiThis article follows Hodge Theory I and Hodge Theory II;
the numbering of the sections here follows that of (which contains to ).In , we gave an introduction to the yoga that underlies and the present article .
However, and are logically independent of ;
they also do not contain all the results that were stated in .In , we introduced the Hodge theory of non-singular (not necessarily complete) algebraic varieties.
Here we treat the case of arbitrary singularities.
From [hodge-theory-iii-7 (?)] onwards, we will make essential use of results from to .In the case of a complete singular algebraic variety , the fundamental idea is to use the resolution of singularities to replace by a simplicial system of smooth projective schemes
(or, as we will say, by a smooth projective simplicial scheme).
The results of [hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] allow us to find such simplicial schemes that have, in a suitable sense, the same cohomology as .
We can then "express", via a spectral sequence, the cohomology of in terms of the cohomology of the .
Following the general principles of , we can then, on each , use classical Hodge theory, and obtain an induced mixed Hodge structure on the cohomology of .In the case of an arbitrary algebraic variety , we "replace" by a smooth simplicial scheme with compactification given by a smooth projective simplicial scheme .
We can further arrange things so that is a normal crossing divisor, given by a union of smooth divisors .
We thus "express", via a spectral sequence, the cohomology of in terms of the cohomology of the -fold intersections of the (for ), and we obtain an induced mixed Hodge structure on .[hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] constitute a summary (without proofs) of the theory of "cohomological descent".
This theory, in the more general framework of a topos, is laid out in [SD].In [hodge-theory-iii-7.1 (?)], we restate certain results obtained previously, but in the language of filtered derived categories;
[hodge-theory-iii-7.2 (?)] contains a simpler proof of the two-filtrations lemma than that of .The unappealing [hodge-theory-iii-8.1 (?)] is the heart of this work.
The essential result is [hodge-theory-iii-8.1.15 (?)].
In its proof, to be able to apply the two-filtrations lemma, we use the fact that every morphism of mixed Hodge structures is strictly compatible with the filtrations and ().
In [hodge-theory-iii-8.2 (?)], we define the mixed Hodge structure of (for a separated scheme).
We show that the Hodge numbers of can only be non-zero for .
For \dim X]]>, we have a more precise result ([hodge-theory-iii-8.2.4 (?)]).
If is smooth (resp. complete), they can only be non-zero if, further, (resp. ).
If is a complete subscheme of , with complement , with complete and smooth of dimension , then we can, by N. Katz, interpret this "duality" between the complete case and the smooth case as coming from the "Alexander duality"
From the functorial properties of the theory thus constructed, we obtain the useful corollaries [hodge-theory-iii-8.2.5 (?)] to [hodge-theory-iii-8.2.8 (?)], which we state in terms independent of any Hodge theory.In [hodge-theory-iii-8.3 (?)], we endow the cohomology of any simplicial scheme with a mixed Hodge structure.
This generality is not illusory.
We can interpret relative cohomology spaces as being the cohomology of suitable simplicial schemes.
Let be the classifying space of the underlying Lie group of an algebraic group .
We can interpret the cohomology of as being the cohomology of a suitable simplicial scheme.
In [hodge-theory-iii-9.1 (?)], after having calculated the mixed Hodge structure on the cohomology of (for linear), we thus obtain that of .
As a corollary, we find that, if a linear group acts on a non-singular complete variety , then the map
factors through .In [hodge-theory-iii-10 (?)], we interpret in terms of abelian schemes the mixed Hodge structures of pure degree (see [hodge-theory-iii-0.5 (?)]), and we consider in detail the of curves.The theory developed up until here is an absolute theory (we do not study the functors ), and deals only with constant coefficients.
I conjecture that, if is a variation of polarisable Hodge structures (in the sense of Griffiths [G1970a]) on a scheme , then the cohomology of with coefficients in the local system is endowed with a natural mixed Hodge structure.
I can only prove this when is complete.