Lowest Weights in Cohomology of Variations of Hodge Structure (II)
aa r X i v : . [ m a t h . AG ] D ec Lowest Weights in Cohomology of Variations ofHodge Structure ∗ Chris PETERSDepartment of Mathematics, University of GrenobleUMR 5582 CNRS-UJF, 38402-Saint-Martin d’H`eres, France [email protected] andMorihiko SAITORIMS Kyoto University, Kyoto 606-8502 Japan [email protected]
December 10, 2008
Abstract
Let X be an irreducible complex analytic space with j : U ֒ → X animmersion of a smooth Zariski open subset, and let V be a variationof Hodge structure of weight n over U . Assume X is compact K¨ahler.Then provided the local monodromy operators at infinity are quasi-unipotent, IH k ( X, V ) is known to carry a pure Hodge structure ofweight k + n , while H k ( U, V ) carries a mixed Hodge structure of weight ≥ k + n . In this note it is shown that the image of the natural map IH k ( X, V ) → H k ( U, V ) is the lowest weight part of this mixed Hodgestructure. In the algebraic case this easily follows from the formalism ofmixed sheaves, but the analytic case is rather complicated, in particularwhen the complement X − U is not a hypersurface. Introduction
For a compact K¨ahler manifold X the decomposition of complex valued C ∞ differential k -forms into types induces the Hodge decomposition for the deRham group H k ( X, C ) equipping this group with a pure weight k Hodgestructure. For singular or non-compact complex analytic spaces this is nolonger true in general. For instance H ( C ∗ ) has rank 1 while it should haveeven rank if it would carry a weight 1 Hodge structure.Cohomology groups of algebraic varieties instead carry a canonical mixedHodge structure , i.e. there is a rationally defined increasing weight filtration ∗ MSC2000 classification: 14C30, 32S35
1o that the k -th graded pieces carry a weight k Hodge structure. In theabove example there is only one weight, namely 2 and H ( C ∗ ) is pure ofweight (1 , U the weight filtration can be seen on the levelof forms as follows. First choose a so-called good compactification, i.e. asmooth (projective) compactification X such that D = X − U is a divisorwith normal crossings. De Rham cohomology of U is the cohomology ofthe full complex of smooth forms on U but it can also be calculated usingthe subcomplex of rational forms having at most logarithmic poles along D , and the weight filtration is given by the number of logarithmic poles.Indeed, H k ( U ) carries a mixed Hodge structure with W k − H k ( U ) = 0 andwhere W k H k ( U ) is the image of the restriction H k ( X ) → H k ( U ). In theanalytic case, a similar assertion holds provided a K¨ahler compactification X of U exists. All of these assertions are well known consequences of Deligne’stheory.In the analytic category we work with manifolds U Zariski-open in somecompact K¨ahler space X . For these, good compactifications exist as in the al-gebraic case . The weight filtration of the mixed Hodge structure on H k ( U )may (and indeed does) depend on the compactification as shown by thefollowing example. Example. U = C ∗ × C ∗ can be analytically compactified in two ways: one is X = P × P , a second one is the compactification Y which is the total spaceof the P -bundle on an elliptic curve E associated to the non-trivial extensionof a trivial line bundle by a trivial line bundle. See [P-S, Example 4.19]. Thefirst has W H ( U ) = 0 while the second has W H ( U ) ≃ H ( E ). Deligne’sresults imply that this would not happen if X and Y can be dominated bya third smooth projective compactification: indeed, two birationally equiv-alent compactifications would give the same mixed Hodge structures.In the analytic category we are thus led to introduce the notion of bimero-morphic equivalence: two smooth K¨ahler compactifications X and Y of U are said to be bimeromorphically equivalent if they are dominated by athird smooth K¨ahler compactification Z of U . If, moreover, the dominatingbimeromorphic morphisms Z → X and Z → Y are projective, we say that X and Y are projective-bimeromorphically equivalent . So U always has a goodcompactification projectively bimeromorphically equivalent to X , but theremay be other good compactifications which are not even bimeromorphicallyequivalent to X as our example shows. However, Deligne’s results implythat for any good K¨ahler compactification Z of U we have that W k H k ( U )is the image of H k ( Z ). Hence, in our example, one still has that W H ( U )is the image of H of the compactification in both cases. More details can be found in § f : Y → X between compact K¨ahler spaces; these involve theterms H q ( X, R p f ∗ Q Y ). Assuming that there is a non-empty Zariski-opensubset U ⊂ X over which Y and f are smooth, the sheaf R p f ∗ Q Y | U isindeed locally constant and its fibers carry a weight p Hodge structure.In fact these can be assembled to give the prototype of what is called a variation of weight p Hodge structure (cf. for instance [C-S-P]). So it isnatural to look at H k ( U, V ) where V is a local system. The replacement for H k ( X ) is intersection cohomology IH k ( X, V ), and there is an intrinsic wayto relate this to ordinary cohomology. Indeed, the adjunction morphismgives a canonical map IH k ( X, V ) → H k ( U, V ) (see Remark 1.2 for details).As is the case for R p f ∗ Q Y | U , one assumes that V carries a variation ofHodge structure. An extra technical assumption on V has to be made whichis known to hold for R p f ∗ Q Y | U : we say that V is quasi-unipotent at infinitywith respect to X if for some (or any) embedded resolution X ′ of ( X, X − U )the local monodromy operators around the branches of X ′ − U are quasi-unipotent. Indeed, if V is quasi-unipotent at infinity with respect to X wehave canonical (pure , respectively mixed) Hodge structures on IH k ( X, V ),respectively H k ( U, V ). Moreover, the mixed Hodge structure on H k ( U, V )depends only the projective bimeromorphic equivalence class of X . This willbe recalled in §
3. See in particular Coroll. 3.5.To motivate the statement of the main theorem below, recall Zucker’sconstruction [Zuc] for dim X = 1. Let j : U ֒ → X be the embedding of U into its compactification. The sheaf j ∗ V is quasi-isomorphic to the complexof holomorphic forms with values in V and with L growth conditions atthe boundary (with respect to the Poincar´e metric). Forgetting the growthconditions gives a complex which computes the cohomology of V on U ;whence a natural restriction map L H k ( U, V ) → H k ( U, V ). The source isnothing but another incarnation of IH k ( X, V ) (Remark 3.6) and indeed,one of the main results from [Zuc] states that it has a pure Hodge structure;moreover, it maps to the lowest weight part of the (special case of our) mixedHodge structure on the target. Hence, in this setting, the lowest weight“comes from the compactification”.The main result of this note concerns a generalization of the lowest weightproperty containing both Zucker’s result and the constant coefficients caseas special cases: Theorem.
Assume U is a smooth complex manifold, j : U ֒ → X ananalytic-Zariski open inclusion into a compact K¨ahler space and let V bea local system on U , quasi-unipotent at infinity with respect to X and car- Because this system is defined over Z , see [Schm, Lemma 4.5]. its weight is k + n where n is the weight of the variation of Hodge structure on V . ying a polarizable variation of Hodge structure. Then a) the natural morphism IH k ( X, V ) → H k ( U, V ) ( ∗ ) is a morphism of mixed Hodge structures; b) the image of the map in a) is exactly the lowest weight part of H k ( U, V ) and is the same for K¨ahler compactifications which are projective-bimero-morphically equivalent. Let us make some comments on the statement of the theorem and itsproof. Note that X is not assumed to be a smooth compactification andthat X − U need not be a divisor with normal crossings. A condition like X being K¨ahler is however unavoidable. As to the proof, a first point thatneeds to be shown is that the natural map (*) preserves Hodge and weightfiltrations. The second point is that the image of this map, which thenlands into the lowest weight part, is exactly the lowest weight part. Finally,since as we have seen, the mixed Hodge structure on H k ( U, V ) depends onthe compactification, one would like to show the lowest weight part dependsonly on the bimeromorphic equivalence class, as in the case of constant coef-ficients. We can show this only for projective-bimeromorphically equivalentcompactifications. The reason is that the decomposition theorem in the an-alytic setting at the moment is only available for projective morphisms (see[Sa88, 5.3.1]).The argument is not too hard if the complement is a hypersurface. Thiscase is treated in § H k ( U, V ). To see how it relates to the Hodge structure on IH k ( X, V ) oneneeds a theory of mixed Hodge complexes on analytic spaces developed in § § l -adic situation and for constant coefficients (actuallythis works as long as the formalism of mixed sheaves [Sa91] is satisfied).Note also that our main theorem in the algebraic case does not follow fromthe mixed Hodge version of [Mor, 3.1.4] unless j is an affine morphism sincethe t -structure in loc. cit. is not associated to the mixed complexes of weight ≤ k in the usual sense, see [Mor, 3.1.2] (and Remark 3.9 below).The first named author wants to thank Stefan M¨uller-Stach for askingthis question and urging him to write down a proof.4 Perverse sheaves
We only give a minimal exposition of the theory of perverse sheaves toexplain the properties which will be used below. We shall only be workingwith the so-called middle perversity which respects Poincar´e duality. Fulldetails can be found in [B-B-D].Let X be a complex analytic space. The category of perverse “sheaves”of Q -vector spaces on X , denoted by Perv( X ; Q ), is an abelian category . Thefact that it is abelian follows from its very construction as a core with respectto a t -structure. While the details of this are not so relevant for what follows,one needs to know that the starting point is formed by the constructiblesheaves of Q -vector spaces on X . By definition these are sheaves of finitedimensional Q -vector spaces which are locally constant on the strata of someanalytic stratification of X . We assume that the stratification is algebraic in the algebraic case. The simplest examples of such sheaves are the locallyconstant sheaves on X itself, or those which are locally constant on somelocally Zariski closed subset Z of X but zero elsewhere.A core is defined with respect to a so-called t -structure and in the per-verse situation the t -structure is defined by certain cohomological conditions,the so called support and co-support conditions. Indeed, instead of startingfrom complexes of constructible sheaves on X one departs from D b c ( X ; Q ) : the derived category of bounded complexesof sheaves of Q -vector spaces on X withconstructible cohomology sheaves. (1)By definition a perverse sheaf is such a complex F which obeys the supportand co-support conditions:dim supp H p ( F ) ≤ − p, dim supp H p ( D F ) ≤ − p, where D F := R Hom( F, D X ) is the Verdier dual of F and D X is the dualizingcomplex. For X smooth and d -dimensional, the case we shall be interestedin, D X is just Q X ( d )[2 d ]. The support condition implies that H p ( F ) = 0for p > H p ( F ) = 0 for p < − d (where d = dim X ): perverse sheaves are complexes “concentrated in degreesbetween − d and 0”.On a complex manifold a (finite rank) local system of Q -vector spaces V can be made perverse by placing it in degree − d : the complex V [ d ] is aperverse sheaf. If X is no longer smooth this complex has to be replaced bythe so-called intersection complex. Indeed, if U ⊂ X is a dense Zariski-opensubset of X which consists of smooth points and V is any (finite rank) localsystem of Q -vector spaces on U the intersection complex IC X ( V [ d ]) can beconstructed as in [B-B-D] (and 1.1 below). (It is also called the minimal Some people write IC X ( V ) instead of IC X ( V [ d ]). IH k ( X, V [ d ]) := H k ( IC X ( V [ d ])) . (2) Remark.
Even if X itself is smooth an intersection complex on X need not beof the form e V [ d ] for some local system e V defined on X because of non-trivialmonodromy “around infinity” X − U .The following two results explain the role of these intersection complexes. Theorem 1.1 ([Bor84, Chap.V, 4]) . Let X be a d -dimensional irreduciblecomplex analytic space and let U be a smooth dense Zariski-open subset of X on which there is a local system V of finite dimensional Q vector spaces.The intersection complex IC X ( V [ d ]) is up to an isomorphism in the derivedcategory the unique complex of sheaves of Q -vector spaces on X which isperverse on X , which restricts over U to V [ d ] and which has no non-trivialperverse sub or quotient objects supported on X − U .Remark . In the situation of Theorem 1.1, let j : U ֒ → X be the inclusionand let I = IC X ( V [ d ]). The adjunction morphism j : I → j ∗ j ∗ I induces ahomomorphism H k j : IH k ( X, V ) → H k ( U, V ) (3)which will be used to compare intersection and ordinary cohomology. Theorem 1.3 ([B-B-D]) . If X is compact or algebraic, Perv( X ; Q ) is Ar-tinian and Noetherian. Its simple objects are the intersection complexes F = IC Z ( V [dim Z ]) supported on an irreducible subspace Z ⊂ X and where V is associated to an irreducible representation of π ( U ) , U ⊂ Z the largestopen subset of Z over which F is locally constant. We also need filtered objects in the abelian category Perv( X ; Q ). A priorithese are not represented by filtered complexes in the usual sense of the word,since the morphisms are in a derived category: they are “fractions” [ f ] / [ s ] : K → L where the bracket stands for the corresponding homotopy class, f : K → N is a morphism of complexes and N s ←− L is a quasi-isomorphism.However, the category of sheaves on X with constructible cohomology hasenough injectives and replacing L by a complex L ′ of injective objects, thequasi-isomorphism s becomes invertible up to homotopy and so [ f ] / [ s ] canbe represented by a true morphism K → L ′ . Next, recall: Lemma 1.4.
For any morphism of complexes v : A → B , the morphism inthe derived category defined by it can be represented by an injective morphism A → B ′ of complexes.Proof : Take B ′ := Cone( − id ⊕ v : A → A ⊕ B ). Then A is a subcomplexof B ′ and we get an injective morphism A → B ′ which is identified with v by the quasi-isomorphism (0 , v, id) : B ′ → B .6 orollary 1.5. Let K ∈ Perv( X ; Q ) . Any finite filtration on K can berepresented by a filtered complex in Perv( X ; Q ) .Proof : Induction on the length of the filtration, assumed to be an increasingfiltration W . The above discussion shows that the morphism W i → W i +1 in Perv( X ; Q ) can be represented by a morphism of complexes to whichLemma. 1.4 can be applied. In this section we put together some properties of mixed Hodge moduleswhich will be used in the sequel. These properties are proven in [Sa88] and[Sa90]. See also the exposition [P-S, Cha. 14] where mixed Hodge Modulesare introduced axiomatically.Let X be a complex algebraic variety or a complex analytic space. Thereexists an abelian category MHM ( X ), the category of mixed Hodge modules on X . Remark.
Note that for nonproper complex algebraic varieties X we alwayshave MHM ( X ) = MHM ( X an ) because of the difference between algebraicand analytic stratifications. Note also that a mixed Hodge module on analgebraic variety is always assumed to be extendable under an open im-mersion. The last property cannot be well-formulated in the analytic casedue to the defect of the Zariski topology on analytic spaces, e.g. Zariski-open immersions are not stable by composition and closed subspaces are notintersections of hypersurfaces Zariski-locally. Properties 2.1.
A) There is a functorrat X : D b MHM ( X ) → D b c ( X ; Q ) . (4)such that MHM ( X ) is sent to Perv( X ; Q ). One says that rat X M is theunderlying rational perverse sheaf of M . Moreover, we say that M ∈ MHM ( X ) is supported on Z ⇐⇒ rat X M is supported on Z. B) The category of mixed Hodge modules supported on a point is the cat-egory of graded polarizable rational mixed Hodge structures; the functor“rat” associates to the mixed Hodge structure the underlying rationalvector space.C) Each object M in MHM ( X ) admits a weight filtration W such that • morphisms preserve the weight filtration strictly; • the object Gr Wk M is semisimple in MHM ( X );7 if X is a point the W -filtration is the usual weight filtration for themixed Hodge structure.Since MHM ( X ) is an abelian category, the cohomology groups of any com-plex of mixed Hodge modules on X are again mixed Hodge modules on X . With this in mind, we say that for a complex M ∈ D b MHM ( X ) the weight satisfiesweight[ M ] (cid:26) ≤ n, ≥ n ⇐⇒ Gr Wk H i ( M ) = 0 (cid:26) for k > i + n for k < i + n. We observe that if we consider the weight filtration on the mixed Hodgemodules which constitute a complex M ∈ D b MHM ( X ) of mixed Hodgemodules we get a filtered complex in this category.D) (i) For each morphism f : X → Y between complex algebraic varieties ,there are induced functors f ∗ , f ! : D b MHM ( X ) → D b MHM ( Y ) and f ∗ , f ! : D b MHM ( Y ) → D b MHM ( X ) which lift the functors Rf ∗ , f ! and f − , f ! respectively; the latter two functors are defined on the level of complexesof sheaves on Y (whose cohomology is constructible.)D)(ii) In the analytic case (i) is no longer necessarily true but we have:— for f : X → Y projective or if X is compact K¨ahler and Y = pt, thereare cohomological functors H i f ∗ = H i f ! : MHM ( X ) → MHM ( Y ) whichlift the perverse cohomological functor p R i f ∗ = p R i f ! ;— for any f there are cohomological functors H i f ∗ , H i f ! : MHM ( Y ) → MHM ( X ) which lift p H i f − , p H i f ! respectively;E) The functors f ∗ , f ! do not increase weights in the sense that if M hasweights ≤ n , the same is true for f ∗ M and f ! M .F) The functors f ∗ , f ! do not decrease weights in the sense that if M hasweights ≥ n , the same is true for f ∗ M and f ! M .G) If f is proper, f ∗ preserves weights, i.e. f ∗ neither increases nor de-creases weights. Remarks 2.2.
1) Despite the fact that the functors f ∗ etc. do not exist inthe analytic setting, properties E), F), G) still have a meaning as in [Sa90,2.26] since the weight is defined in terms of cohomology only.2) Since in the analytic setting Zariski-open immersions are not stable bycomposition H i f ∗ M , H i f ! M do not necessarily exist for analytic morphisms f . This explains why in the analytic case D) not all morphisms are allowed.3) The reader may interpret the K¨ahler condition on X in Property D) asthe existence of a projective morphism g from a K¨ahler manifold X ′ onto X . Indeed, the construction of H i f ∗ M for f : X → pt , where M is a pureHodge module, is reduced to the assertion for X ′ : use the decomposition8heorem for g applied to a pure Hodge module on X ′ which is a subquotientof the pullback of M by g . Then it follows from [C-K-S], [K-K86], [K-K87],[K-K89]. For the mixed case we can use the weight spectral sequence.4) It is still unclear whether H i f ∗ M exists for proper K¨ahler morphisms f unless M is constant, see [Sa90b]. If the reader prefers, he may assume thatthe polarizable Hodge modules in this paper are direct factors of the coho-mological direct images of the constant sheaf by smooth K¨ahler morphismsso that the existence of H i f ∗ M follows from the decomposition theorem forthe direct image of the constant sheaf by proper K¨ahler morphisms [Sa90b].The above properties readily imply various basic properties of mixedHodge modules. For example, if M is a complex of mixed Hodge moduleson X its cohomology H q M is a mixed Hodge module on X . Properties B)and D) imply: Lemma 2.3.
Let a X : X → pt be the constant map to the point. Assume X is algebraic or compact K¨ahler. Then for any complex M of mixed Hodgemodules on X H p ( X, M ) := H p (( a X ) ∗ M ) (5) is a mixed Hodge structure. For the proof of the main theorem one needs the following two technicalconstructions. The first is the adjunction construction:
Construction 2.4.
Consider a morphism f : X → Y of algebraic va-rieties and a mixed Hodge module M on Y . The adjunction morphism f : M → f ∗ f ∗ M is a morphism of complexes of mixed Hodge modules.For any bounded complex K of mixed Hodge modules on X , the identity a X = a Y ◦ f induces a canonical identification H n ( Y, f ∗ K ) = H n ( X, K ). Inparticular this holds for K = f ∗ M . Adjunction thus induces a morphism ofmixed Hodge structures H k f : H k ( Y, M ) → H k ( X, f ∗ M ) . (6)In the analytic case this construction remains valid for an open immer-sion j whose complement is a hypersurface (defined locally by a function g ).Indeed, then j ∗ j ∗ M is a mixed Hodge module whose underlying D -modulecomes from localization by g applied to the underlying D -module of M .More generally, consider the complement U of an intersection Z of globalhypersurfaces. Then j ∗ j ∗ M is a complex of mixed Hodge modules due to asecond construction: Construction 2.5 ([Sa90, 2.19, 2.20]) . Let g i , i = 1 , . . . , r be holomorphicfunctions on Y , let Z = T ri =1 g − i (0) red and U = Y − Z . We set Y i = Y − g − i (0) and for I ⊂ { , . . . , r } we set Y I = T i ∈ I Y i . Let i : Z ֒ → Y , j : U ֒ → Y and j I : Y I ֒ → Y be the natural inclusions. Let M be a9ixed Hodge module on Y ; then also j ∗ M , the restriction of M to U , is amixed Hodge module on U and there are quasi-isomorphisms in the category D b MHM ( Y ) i ∗ i ! M ∼ −→ [ · · · → M → B → B · · · B r → , B k = M | I | = k ( j I ) ∗ j ∗ I Mj ∗ j ∗ M ∼ −→ [ · · · → B → B → B · · · → B r → , B k in degree k − Lemma 2.6 ( [Sa90, (4.4.1)]) . Let i : Z ⊂ Y be a closed immersion and j : U = Y − Z ֒ → Y be the inclusion of the complement. Assume Y, Z arealgebraic or, alternatively, that Z is an intersection of global hypersurfacesof Y . Let M be a mixed Hodge module on Y . There is a distinguishedtriangle i ∗ i ! M −−−−→ Mj ∗ j ∗ M ❙❙❙♦ ✓✓✓✴ α [1] (7) in the bounded derived category of mixed Hodge modules lifting the analogoustriangle for complexes with constructible cohomology sheaves. The morphism α induces the adjunction morphism H k j : H k ( Y, M ) → H k ( U, j ∗ M ) for j (see (6) ).Proof : In the algebraic setting the local constructions 2.5 for a suitableaffine cover patch together to give globally defined quasi-isomorphisms for i ∗ i ! M and j ∗ j ∗ M . The local construction shows the existence of the distin-guished triangle. See [Sa90, 4.4.1] for details.The same argument applies in the analytic case under the assumptionthat Z is a global complete intersection. See the proof of [Sa90, 2.19]. Remark . The reader may wonder what happens in the general setting ofanalytic spaces. The problem is that Construction 2.5 can not be globalizedto complexes of mixed Hodge modules. However, the cohomology sheaves ofthe complexes do make sense globally and are indeed mixed Hodge modules.Hence also the long exact sequence in cohomology associated to (7) existsin the category of mixed Hodge modules.
In this section X is an irreducible compact K¨ahler analytic space of dimen-sion d . Let j : U ֒ → X be the inclusion of a dense Zariski-open subset forwhich we make the crucial assumption that U is smooth. We shall write triangles also as M ′ → M → M ′′ → [1].
10e shall not review here the definitions and properties of polarizableHodge modules. For our purposes we only need the following basic resultlinking variations of Hodge structures and polarizable Hodge modules [Sa88,Th. 5.4.3]:
Theorem 3.1.
Suppose that V is a polarizable variation of Hodge structureon U of weight n . If U is smooth, there is a polarizable Hodge module V Hdg of weight n + d on U whose underlying perverse component is V [ d ] . There is however, one important aside to make at this point. In thealgebraic setting mixed Hodge modules are assumed to extend under openimmersions, but this ceases to hold in the analytic category. Instead, onereplaces it by the following condition on the underlying local system.
Definition 3.2.
Let V be a local system on U . We say that V is quasi-unipotent at infinity with respect to X , if for some (or any) choice of anembedded resolution X ′ of ( X, X − U ) the local monodromy operators of V around X ′ − U are quasi-unipotent. Remark.
By [Ksh] this property is independent of the choice of X ′ anddepends only on the bimeromorphic equivalence class of X . By definition X ′ is a good compactification of U together with a bimeromorphic map f : X ′ → X ; these exist: by blowing up X in suitable ideals on can evenassume that f is projective. So one can test quasi-unipotency on such X ′ .For a polarizable Hodge module this notion leads to pure Hodge modules(see [Sa90]). In both settings (polarizable algebraic Hodge modules andpure mixed Hodge modules in the analytic case) one obtains semi-simplecategories: this is implied by the polarizability condition, see also 2.1.C).Both categories satisfy moreover the strict support condition:
Property 3.3.
A polarizable weight n Hodge module M is a direct sumof polarizable weight n Hodge modules M Z which have strict support Z where Z are irreducible subvarieties of X , and the same assertion holds forpure Hodge modules.By [Sa90, 3.20, 3.21] one has: Theorem 3.4.
Assume that U is smooth and that V is a quasi-unipotentat infinity with respect to X and underlies a polarized variation of Hodgestructures of weight n on U . Then there is a unique pure Hodge module V Hdg X of weight n + d on X having strict support in X and which restrictsover U to V Hdg . M is said to have strict support Z if it is supported on Z but no quotient or sub objectof M has support on a proper subvariety of Z . emark. Note that this checks with the assertion in Theorem 1.1 which holdsfor the rational component of the mixed Hodge modules. More precisely:the intersection complex IC X ( V [ d ]) is the rational component of V Hdg X . Thisremark is crucial for the proof of the next corollary. Corollary 3.5. There exists a mixed Hodge structure on H k ( U, V ) . Itdepends only on the projective bimeromorphic equivalence class of X ; IH k ( X, V ) carries a pure Hodge structure of weight k + n .Proof : 1) Replacing X by a suitable blow up, we may assume that X isa good compactification of U . By construction 2.5 j ∗ j ∗ V Hdg X = j ∗ V Hdg then is a mixed Hodge module on X and by Lemma 2.3 the cohomol-ogy group H k ( U, V ) carries the mixed Hodge structure H k − d ( j ∗ V Hdg ). Ifthere are two good compactifications X , X with a projective morphism π : X → X inducing an isomorphism over U , then with j k : U ֒ → X k , k = 1 , Rπ ∗ Rj ∗ V Hdg X = Rj ∗ V Hdg X by the uniqueness of Rj ∗ , see e.g. [Sa90, 2.11].2) Since V Hdg X is a pure Hodge module, by the previous Remark and Lemma 2.3 H k ( X, IC X ( V ) = IH k ( X, V ) carries the mixed Hodge structure H k − d ( X, V
Hdg X ),which by Properties 2.1 G) is pure of weight k + n . Remark . Suppose that in addition X is smooth and X − U is a divisorwith normal crossings. Then, by [C-K-S, Theorem 1.5], [K-K86], [K-K87],[K-K89] IH k ( X, V ) can be identified with L H k ( U, V ) provided one mea-sures integrability with respect to the Poincar´e metric around infinity (one isin the normal crossing situation, so locally around infinity one has a productof disks and punctured disks). Summarizing: H k ( IC X ( V )) = IH k ( X, V ) = L H k ( U, V )has a pure Hodge structure of weight k + n .Next one wants to relate intersection cohomology and ordinary cohomol-ogy. This is the content of the main theorem: Theorem 3.7.
Assume that U is smooth, that V is a quasi-unipotent atinfinity with respect to X and that it carries a polarized variation of Hodgestructure of weight n . Then a) The natural morphism H k j : IH k ( X, V ) → H k ( U, V ) (see (3) ) is a morphism of mixed Hodge structures; b) the image of H k j is exactly the lowest weight part of H k ( U, V ) and thisimage is the same for K¨ahler compactifications which are projective-bimero-morphically equivalent to X . roof in the algebraic case. Let i : Z = X − U ֒ → X be the inclusion. Set M = V Hdg X , M ′ = j ∗ V Hdg = j ∗ j ∗ V Hdg X and M ′′ = i ∗ i ! V Hdg X . Formula (6) forthe inclusion j : U ֒ → X and the mixed Hodge module M := V Hdg X showsthat (3) is indeed a morphism of mixed Hodge structures.Form the distinguished triangle (7). Portion of its associated long exactsequence in hypercohomology reads · · · → IH k ( X, V ) H k j −−−−→ H k ( U, V ) (cid:13)(cid:13) (cid:13)(cid:13) H k − d ( X, M ) −−−−→ H k − d ( X, M ′ ) → H k − d +1 ( X, M ′′ ) → . . . (8)By Theorem 3.4 M = V Hdg X is pure of weight n + d , and so by Property 2.1.Fthe complex i ! V Hdg X has weight ≥ n + d . By Property 2.1.G this also holdsfor the complex M ′′ = i ∗ i ! V Hdg X . Applying once more Property 2.1.G tothe functor ( a X ) ∗ one sees that H k − d +1 ( X, M ′′ ) has weights ≥ k + n + 1and hence the image of the map (3) is exactly the weight ( k + n )-part of H k ( U, V ).In the algebraic category the last assertion of b) can be replaced by astronger assertion: we may assume that two compactifications are relatedby a proper algebraic morphism to which the decomposition theorem canbe applied. Instead of giving full details here we refer to the proof in theanalytic setting which is given at the end of § Strategy of the proof in the analytic setting. If Z is a hypersurface, the sameproof works in view of Lemma 2.6. This Lemma also shows that (3) is amorphism of mixed Hodge structures in this case.In the general situation one has to perform a suitable blow-up π : X ′ → X which is the identity in U and such that Z ′ = X ′ − U is a divisor. Nowwe would like to apply the functor π ∗ . The problem is that this functor doesnot exist in the derived categories of mixed Hodge modules. So we have tofind a substitute for this which still preserves enough of the Properties 2.1so that we can complete the proof as in the algebraic case. It turns out thatthe correct category to use is the one of mixed Hodge complexes. See § § Remark . In the algebraic setting the following claim is easily shown toimply the main result as well and can be seen as a refinement of it.
Claim . Suppose Z is a locally principal divisor or j is an affine morphism.Then the adjunction morphism j : V Hdg X → j ∗ j ∗ V Hdg X is injective andidentifies V Hdg X with the lowest weight part of j ∗ j ∗ V Hdg X = j ∗ V Hdg
Indeed, the extra hypothesis on j implies (see Construction 2.5 and[Sa90, 2.11]) that j ∗ V Hdg is a mixed Hodge module (not just a complex ofmixed Hodge modules) and the main theorem then follows easily from theClaim. The latter follows from the long exact sequence 0 → H i ∗ i ! V Hdg X → Hdg X → j ∗ V Hdg → H i ∗ i ! V Hdg X using that the strict support condition im-plies that H i ∗ i ! V Hdg X = 0.The above claim can alternatively be shown using adjunction. This washow the first named author originally proved the main result . Here isthe argument. It suffices to show that the lowest weight part W d + n M of M = j ∗ V Hdg has no quotient or sub object supported on D = X − U . It isa pure weight mixed Hodge module, and hence, by Property 2.1.C a semi-simple object in the category of mixed Hodge modules. By construction, itrestricts to V Hdg on U . By semi-simplicity a quotient object is also a subobject and hence it suffices to show that there are no mixed Hodge modules N of pure weight n + d supported on D for which Hom A ( N, W n + d M ) = 0in the abelian category A of mixed Hodge modules. Functoriality of theweight filtration implies Hom A ( N, W n + d M ) = Hom A ( N, M ). Let D ( A )be the derived category of bounded complexes in A . Since the naturalmap Hom A ( N, M ) → Hom D ( A ) ( N, M ) is a bijection (see [Verd77, p. 293])it is enough to show that Hom D ( A ) ( N, M ) = 0. In the derived categoryone can use the adjunction for ( j ∗ , j ∗ ) yielding Hom D ( A ) ( N, j ∗ V Hdg ) =Hom D ( A ) ( j ∗ N, V
Hdg ) = 0 since j ∗ N = 0. Remark . It would not be difficult to construct a mixed Hodge version of[Mor, 3.1.4]. However, this would not immediately imply our main theoremunless j is an affine morphism. Indeed, the t -structure in loc. cit. is definedby the condition that p H i K has weight ≤ k and not ≤ i + k as in the caseof mixed Hodge complexes of weight ≤ k , see [Mor, 3.1.2]. It does not seemthat there exists a t -structure associated to mixed complexes of weight ≤ k since the weight filtration is not strict and the weight spectral sequence doesnot degenerate at E (see also Section 5 on the proof of Theorem 3.7 in theanalytic case where mixed Hodge complexes in the Hodge setting are used). For the proof of Theorem 3.7 in the analytic case we need a theory of mixedHodge complexes on analytic spaces which refines Deligne’s theory [Del71]of cohomological mixed Hodge complexes. We present it here in a rathersimplified manner which has the defect that the mapping cones are not well-defined. However, this does not cause a problem for the proof of Theorem 3.7since all we need is the existence of the long exact sequence (12). See [Sa00]for a more elaborate formulation taking care of the problem with the cones.
Notation. — M F W ( D X ): the category of filtered D X -modules ( M, F )with a finite filtration W . For singular X this can be defined by usingclosed embeddings of open subsets of X into complex manifolds, see [Sa88,2.1.20]. See http://arxiv.org/abs/0708.0130v2 D b h F W ( D X ): the derived category of bounded complexes ( M, F, W ) suchthat 1) the sheaves L p H i F p Gr Wk M are coherent over the sheaf L p F p D X and 2) the sheaves H i Gr Wk M are holonomic D X -modules.— D b c W ( X, Q ): the derived category of of bounded filtered complexes ( K, W )such that W is finite and Gr Wk K ∈ D b c ( X, Q ) for any k : we define D b c W ( X, C )similarly.— D b h F W ( D X , Q ): the “fibre” product of D b h F W ( D X ) and D b c W ( X, Q )over D b c W ( X, C ) where the functor DR : D b h F W ( D X ) → D b c W ( X, C ) in-duced by the de Rham functor is used to glue the two categories. Moreprecisely, its objects are triples M = (( M, F, W ) , ( K, W ) , α )where ( M, F, W ) ∈ D b h F W ( D X ), ( K, W ) ∈ D b c W ( X, Q ) and α : DR( M, W ) ∼ = ( K, W ) ⊗ Q C in D b c W ( X, C )and morphisms in the category are pairs of morphisms of D b h F W ( D X ) and D b c W ( X, Q ) compatible with α . Forgetting the filtration W we can define D b h F ( D X ), D b c ( X, Q ) and D b h F ( D X , Q ) similarly.— Gr Wk M = (Gr Wk ( M, F ) , Gr Wk K, Gr Wk α ) ∈ D b h F ( D X , Q ) . Definition 4.1.
1) The category of mixed Hodge complexes
MHC ( X ) is thefull subcategory of D b h F W ( D X , Q ) consisting of M = (( M, F, W ) , ( K, W ) , α )satisfying the following conditions for Gr Wk M for any k, i :(i) The Gr Wk ( M, F ) are strict and we have a decompositionGr Wk M ∼ = M j ( H j Gr Wk M )[ − j ] . (9)(ii) The H i Gr Wk M are polarizable Hodge modules of weight k + i .2) Let MHW ( X ) denote the category of weakly mixed Hodge modules , i.e. itsobjects have a weight filtration W for which the gradeds Gr Wk are polarizableHodge modules of weight k , but there is no condition on the extensionbetween the graded pieces.3) We say that M u → M ′ v → M ′′ w → M [1] is a weakly distinguished triangle in MHC ( X ) if u, v, w are morphisms of MHC ( X ) and its underlying triangleof complexes of sheaves of Q -vector spaces is distinguished. Here the weightfiltration W on M [1] is shifted by 1 so that M [1] is a mixed Hodge complex. Remark.
In the case X =pt , we do not have to assume the decomposition (9)in condition (i) of Definition 4.1,1). One reason is that this is only needed toprove the stability by the direct image under a morphism from X . Anotherreason is that this decomposition actually follows from the other conditionsin this case since the category of vector spaces over a field is semisimple.15e have by [Sa88, 5.1.14] Proposition 4.2.
The category
MHW ( X ) is an abelian category whose mor-phisms are strictly compatible with ( F, W ) . For a mixed Hodge complex M , set H i M = ( H i ( M, F ) , p H i ( K ) , p H i α ) . We put a weight filtration on it by letting W k be the image of H i W k − i M (or, equivalently, the one induced by the filtration Dec W for the underlying D -module (cf. Proposition 4.3 below). This shift of the filtration W comesfrom condition (ii) in the above definition of MHC ( X ).Using [Sa88, 1.3.6 and 5.1.11], etc. we have Proposition 4.3.
With the weight filtration W defined above, the H i M are weakly mixed Hodge modules. There is a weight spectral sequence in theabelian category of weakly mixed Hodge modules MHW ( X ) E p,q = H p + q Gr W − p M ⇒ H p + q M , (10) which degenerates at E , and whose abutting filtration on H p + q M coincideswith the weight filtration of weakly mixed Hodge modules shifted by p + q asabove, i.e. E p,q ∞ = Gr Wq H p + q M (11) Moreover, ( M, F,
Dec W ) is bistrict, and the weight filtration on H p + q M isinduced by Dec W where M is the underlying D -module of M and (Dec W ) k M i := Ker( d : W k − i M i → Gr Wk − i M i +1 ) . Combining this with Proposition 4.2 we get
Proposition 4.4.
A weakly distinguished triangle as in Definition 4.1, 3)induces a long exact sequence in the abelian category
MHW ( X ) → H i M u → H i M ′ v → H i M ′′ w → H i +1 M → . (12)For a morphism of mixed Hodge complexes u : M → M ′ , there is a mapping cone M ′′ := Cone( u : M → M ′ ) in the usual way. Here the weightfiltration W on M [1] is shifted by 1 so that Gr W u in the graded piecesof the differential of M ′′ vanishes and hence conditions (i) and (ii) aboveare satisfied. However, M ′′ is not unique up to a non-canonical isomorphismbecause of a problem of homotopy. So we cannot get a triangulated categoryalthough there is a weakly distinguished triangle M → M ′ → M ′′ → [1]which by Proposition 4.4 induces the long exact sequence (12) in the category MHW ( X ).Since the weight filtration on the perverse component of a weakly mixedHodge module can be represented by an honest filtered complex (Cor. 1.5)we have: 16 roposition 4.5. Considering a weakly mixed Hodge module as a mixedHodge complex concentrated in degree we get a functor ι X : MHW ( X ) → MHC ( X ) . Let f : X → Y be a projective morphism, and let M be a polarizableHodge module. The object f ∗ ( ι X ( M )) belongs to D b h F ( D Y , Q ). The decom-position theorem [Sa88, 5.3.1] can be applied to M and applying ι Y to theresulting Hodge modules yields elements in D b h F ( D Y , Q ). The uniquenessof the decomposition [Del94] then implies: Theorem 4.6.
Let f : X → Y be a projective morphism, and M be theimage of a polarizable Hodge module by ι X . Then we have a decomposition f ∗ M ∼ = M i ( H i f ∗ M )[ − i ] in D b h F ( D Y , Q ) . Combining this with Properties 2.1.D) (ii) we get
Corollary 4.7.
Mixed Hodge complexes and weakly distinguished trianglesare stable by the direct image under f : X → Y if f is projective or if X iscompact K¨ahler and Y = pt .Remark. Note that the stability by direct images asserted in Corollary 4.7does not follow from Theorem 4.6 if we replace k + i by k in condition (ii)in the above definition of MHC ( X ). (This causes the shift of the filtration W in Proposition 4.3 below.) Let π : X ′ → X be a bimeromorphic projective morphism inducing theidentity over U and such that X ′ − U is a hypersurface (defined locally by afunction). Let j ′ : U → X ′ denote the inclusion. Then R j ′∗ V [ d ] is a perversesheaf, and underlies a mixed Hodge module j ′∗ V Hdg , see [Sa90, 2.17]. ByProposition 4.5 this gives a mixed Hodge complex concentrated in degree 0 M ′ = (( M ′ , F, W ) , ( K ′ , W ) , α ) := ι X ′ ( j ′∗ V Hdg ) , (13)such that K ′ = R j ′∗ V [ d ] and M ′ | U is identified with V Hdg . We denote thedirect image of M ′ by M = (( M, F, W ) , ( K, W ) , α ) := π ∗ M ′ = ( π ∗ ( M ′ , F, W ) , π ∗ ( K ′ , W ) , π ∗ α ) . By Corollary 4.7 this is a mixed Hodge complex since π is projective. Proposition 5.1.
We have Gr Wd + n H M = ι X ( V Hdg X ) , and Gr Wk H i M = 0 if k = d + n + i, i = 0 or if k < d + n + i . roof : It suffices to show the assertion for the underlying complex of D -modules M since the condition on strict support in Theorem 3.4 is detectedby its underlying D -module. Moreover we may restrict to a sufficiently smallopen subset Y of X enabling us to apply Construction 2.5.So let g , . . . , g r be functions on Y such that Z ∩ Y = T i g − i (0). Set Y i = Y − g − i (0). Abusing notation, let i : Y ∩ Z → Y , j : Y − Z → Y denote the inclusions. By Lemma 2.6 there is a distinguished triangle i ∗ i ! ( V Hdg X | Y ) → V Hdg X | Y → j ∗ j ∗ ( V Hdg X | Y ) → [1] , inducing a long exact sequence of cohomology. Claim . The underlying bifiltered D -modules of ι Y ( H i j ∗ j ∗ ( V Hdg X | Y )) and H i M| Y are isomorphic to each other.Suppose that the Claim has been shown. Then the same argument asin the proof of Theorem 3.7 in the algebraic case proves the result of theProposition. Indeed, we have the exact sequence H i ( V Hdg X | Y ) → H i j ∗ j ∗ ( V Hdg X | Y ) → H i +1 i ∗ i ! ( V Hdg X | Y ) , and H i +1 i ∗ i ! ( V Hdg X | Y ) has weights ≥ d + n + i + 1 by Properties 2.1.F and G.This gives the assertion for i = 0 since V Hdg X | Y is pure of weight d + n . For i = 0 we have H i ( V Hdg X | Y ) = 0 and hence the last morphism of the exactsequence is injective so that the assertion follows. Proof of the Claim.
Let Y ′ = π − ( Y ), Y ′ i = π − ( Y i ), and g ′ i = π ∗ g i . ByConstruction 2.5 the associated ˇCech complex gives a resolution of j ′∗ V Hdg .The components of this ˇCech complex are direct sums of ( j ′ I ) ∗ ( V Hdg | Y ′ I )where Y ′ I = T i ∈ I Y ′ i with the inclusion j ′ I : Y ′ I → Y . By the uniqueness ofthe open direct image in [Sa90, 2.11] we have moreover π ∗ ( j ′ I ) ∗ ( V Hdg | Y ′ I ) = ( j I ) ∗ ( V Hdg | Y I ) , where j I : Y I := T i ∈ I Y i → Y . So we get the desired isomorphism (usingthe filtration Dec W from Proposition 4.3), and Proposition 5.1 follows.We return to the proof of Theorem 3.7 in the analytic case. ApplyingProposition 5.1 to M ′ , we getGr Wd + n M ′ = ι X ′ (Gr Wd + n j ′∗ V Hdg ) = ι X ′ ( V Hdg X ′ ) . This implies that we get a morphism u ′ : ι X ′ ( V Hdg X ′ ) → M ′ to which weapply π ∗ . The decomposition Theorem 4.6 together with the semisimplicityof polarizable Hodge modules imply that V Hdg X is a direct factor of π ∗ V Hdg X ′ .So we get a morphism u : ι X ( V Hdg X ) → M .
18t is not clear whether u is uniquely defined (since the decomposition isnot unique). However, its underlying morphism of Q -complexes coincideswith the canonically defined adjunction morphism j so that it induces thedesired morphism of mixed Hodge structures H i j : IH i ( X, V ) → H i ( U, V ) . Let M ′′ be a mapping cone of u : ι X ( V Hdg X ) → M as defined in Section 4.Remember (13) that M comes from j ′∗ V Hdg , a mixed Hodge module ofweight ≥ n + d (by Properties 2.1. F)) and hence Gr Wk M = 0 for k < d + n .Then, by definition of the cone, one hasGr Wk M ′′ = Gr Wk M = 0 for k < d + n. (14)Using Proposition 5.1 (e.g. ι X ( V Hdg X ) = Gr Wd + n H M ) together with the longexact sequence (12) we get moreoverGr Wk H i M ′′ = 0 for k ≤ i + d + n. Since by (11) we have E i,k ∞ = Gr Wk H i + k M ′ , the weight spectral sequence(10) implies the surjectivity of E − d − n − ,d + n +1+ j d −−→ E − d − n,d + n + j +11 (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) H j Gr Wd + n +1 M ′′ → H j +1 Gr Wd + n M ′ for all j , and this map splits by the semisimplicity of polarizable Hodgemodules. So we get the surjectivity of H i ( a X ) ∗ H j Gr Wd + n +1 M ′′ → H i ( a X ) ∗ H j +1 Gr Wd + n M ′′ for any i, j. Claim . This implies the surjectivity of H i ( a X ) ∗ Gr Wd + n +1 M ′′ → H i +1 ( a X ) ∗ Gr Wd + n M ′′ for any i. Proof of the claim.
The truncation τ ≤ j on Gr Wk M ′′ splits by the definitionof mixed Hodge complexes so that Gr Wk M ′′ ≃ L i H i (Gr Wk M ′′ )[ − i ] where M ′′ is the underlying D -module of M ′′ . Now the truncation induces a filtra-tion τ ′ on H i ( a X ) ∗ Gr Wk M ′′ and the preceding splitting for Gr Wk M ′′ comingfrom the truncation induces a splitting for H i ( a X ) ∗ Gr Wd + n +1 M ′′ comingfrom τ ′ . Its factors are isomorphic to H i − j ( a X ) ∗ H j Gr Wd + n +1 M ′′ and thisfactor maps surjectively to the factor of H i +1 ( a X ) ∗ Gr Wd + n isomorphic to H i − j ( a X ) ∗ H j +1 Gr Wd + n M ′′ . 19gain using Gr Wk M ′′ = 0 for k < d + n (14) it follows from the weightspectral sequence for ( a X ) ∗ M ′′ thatGr Wk H i ( a X ) ∗ M ′′ = 0 for k ≤ d + n + i. (15)The long exact sequence (12) for the direct image of the weakly distinguishedtriangle of the cone for u under a X : X → pt reads · · · H i ( a X ) ∗ M ′′ → IH i ( X, V ) H i ( j ) −−−−−→ H i ( U, V ) → H i +1 ( a X ) ∗ M ′′ . From Corollary 4.7) this is a sequence of mixed Hodge structures and (15)shows the assertion about the lowest weights.To complete the proof of Theorem 3.7 in the analytic case, we only haveto show the independence of the compactification. The map (3) induced by j is obtained from the natural map of Q -complexes ι : IC X ( V [ d ]) → Rj ∗ V [ d ]after applying the global section functor. We may assume that we have asecond compactification j ′ : U ֒ → Y related to j : U ֒ → X by a projectivemorphism f : X → Y . The map induced by j ′ is similarly obtained fromthe natural map ι ′ : Rf ∗ IC X ( V [ d ]) → Rf ∗ Rj ∗ V [ d ] = ( Rj ′ ) ∗ V [ d ]. As before,the decomposition Theorem 4.6 combined with the fact that polarizableHodge modules form a semi-simplicial category implies that the intersectioncomplex IC Y ( V [ d ]) is a direct factor of Rf ∗ IC X ( V [ d ]) (but the latter mightcontain other direct factors). The restriction of ι ′ to IC Y ( V [ d ]) is exactlyequal to ι while the other direct factors are in the kernel of ι ′ since thesemust be supported on Y − U . It follows that the image of ι does not dependon the compactification and hence neither does the image of (3). References [B-B-D] Beilinson, A., J. Bernstein and P. Deligne: Faisceaux pervers,in:
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