Mixed Hodge complexes and higher extensions of mixed Hodge modules on algebraic varieties

Mixed Hodge complexes and higher extensions of mixed Hodge modules on algebraic varieties

FLORIANIVORRA(*)

ABSTRACT- In [1, 2] A. BeõÆlinson provides a description of the bounded derived category of polarizable mixed Hodge structures in terms of the more flexible triangulated category of polarizable mixed Hodge complexes. In this work we provide a partial generalization of this result to higher dimension.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 14D07, 14F10, 14F40, 14F42.

KEYWORDS. Mixed Hodge modules, mixed Hodge complexes.

1. Introduction

1.1 ± Some background

1.1.1 ± In P. Deligne's Hodge theory [5, 8.1], mixed Hodge complexes were introduced as an auxiliary tool to construct mixed Hodge structures on cohomology groups of algebraic complex varieties. This tool was refined by A. BeõÆlinson in [1, 2] where it plays a crucial role in the definition of absolute Hodge cohomology, a Hodge theoretic precursor of motivic cohomology.

More precisely, A. BeõÆlinson introduces a triangulated category of (polariz- able) mixed Hodge complexes D_H^{b p}and shows that the triangulated functor

D^b(MHS^pQ)!D_H^{b p} (1:1:1 1)

is an equivalence of categories. Then he explains [1, § 4] that Deligne's con- struction provides a mixed Hodge complex RG(X;Q) which lifts the direct imageRpQ_Xin Betti cohomology, and, using the equivalence (1.1.1 1), he defines the absolute (polarizable) Hodge cohomology as the bigraded groups:

H_H^p p(X;Q(q)):ˆHom_D^b_(MHS^p

Q)(Q(0);RG(X;Q)(q)[p]):

(*) Indirizzo dell'A.: Institut de Recherche MatheÂmatique de Rennes, UMR 6625 duCNRS,UniversiteÂdeRennes1,CampusdeBeaulieu,35042RennesCedex(France)

E-mail: florian.ivorra@univ-rennes1.fr

At the time, RG(X;Q) was a substitute for the direct imagepQ^H_X that a not yet defined theory of Hodge coefficients with a formalism of the six op- erations would have provided.

1.1.2 ± Even with the advent of M. Saito's theory of mixed Hodge modules [17, 19], which allows in particular the definition of absolute Hodge cohomology without mentioning the category of mixed Hodge com- plexes (¹), BeõÆlinson's equivalence still remains a useful description of the bounded derived category of polarizable mixed Hodge structures. Indeed in comparison with complexes of polarizable mixed Hodge structures, mixed Hodge complexes are far more flexible objects and much easier to construct, and the equivalence (1.1.1 1) turns out to be a key ingredient for constructing realization functors from triangulated categories of motives to the bounded derived category of polarizable mixedQ-Hodge structures MHS^pQ: in both the construction given by A. Huber in [7, 8] for V. Voevodsky's geometrical motives or by M. Levine [14, Part I,V.2.3.10]

for his own triangulated category of motives the target category of the realization is rather the bounded triangulated category of polarizable mixed Hodge complexes D^b_H^p.

1.1.3 ± For higher dimensional complex algebraic varieties, categories of mixed Q-Hodge complexes were introduced by M. Saito in [20].

More precisely, for every algebraic variety X, he constructs two triangulated categories D_H^b(X;Q)_D and D_H^b(X;Q) together with trian- gulated functors:

where MHM(X;Q) is the abelian category of algebraic mixedQ-Hodge modules overX. ForXˆSpec(C) the categories MHM(SpecC;Q) and MHS^p_Q are equivalent, M. Saito shows [20, 2.10] that the functor e in that case

e:D^b(MHM(SpecC;Q))!D_H^b(SpecC;Q)_D

(¹) M. Saito's theory provides a definition ofpQ^H_X as an object in D^b(MHS^pQ) and one can shows at least forXsmooth that its image by the equivalence (1.1.1 1) is isomorphic to BeõÆlinson's RG(X;Q).

is an equivalence and in particular Saito's category of mixed Hodge complexes is equivalent to D_H^{b p}. Though their definitions are very sim- ilar, Saito's construction is slightly more complicated than BeõÆlinson's construction and it is not clear a priori that they yield equivalent results.

1.1.4 ± The motivation behind these constructions is the comparison between Hodge structures provided by the theory of mixed Hodge modules and Hodge structures provided by the simplicial methods inherited from Deligne's original approach [5]. As simplicial constructions do not work in the framework of mixed Hodge modules, due to the nonexactness of pullback maps for the perverse t-structure, mixed Hodge complexes appears to provide a bridge between the theory elaborated by Saito and other constructions with a simplicial flavor.

1.2 ± Statement of the main result

1.2.1 ± In this paper, for technical reasons, we will have to work with the version of the functoreconstructed in [9]:

D^b(MHM(X;Q))ƒƒƒ!^realX D_H^b(X;Q)_D: (1:2:1 1)

Though the definition of the category of mixed Hodge complexes DH^b(X;Q)_D is too crude for (1.2.1 1) to be essentially surjective for an arbitrary algebraic variety, if one is interested in a higher dimensional generalization of BeõÆlinson's equivalence, the following question arises:

1.2.2 ± QUESTION.Let X be a smooth projective algebraic variety. Is the realization functor(1.2.1 1)fully faithful ? In other words is the morphism (1:2:2 1) Hom_Db(MHM(X;Q))(M;N )!Hom_Db

H(X;Q)D(real_X(M);real_X(N )) an isomorphism for any given complexes of mixed Hodge modulesM;N ? 1.2.3 ± The aim of this work is to provide a partial positive answer to this question for every smooth projective variety. Remark that the realization functor (1.2.1 1) is a Hodge theoretic lifting of the Betti functor real_X;Q constructed in [4, 3.1],i.e.one has a commutative square (up to an invertible 2-isomorphism)

(1:2:3 1)

where D^b_c(X;Q) is the bounded derived category of sheaves ofQ-vector spaces onX^anwith algebraically constructible cohomology and Perv(X;Q) is the abelian category of perverse sheaves inside it. As shown by BeõÆlinson in [3] the functor real_X;Qis an equivalence of categories. Therefore a pos- itive answer to 1.2.2 can be seen both as a higher dimensional general- isation of the equivalence in [1] and a Hodge theoretic version of the equivalence in [3].

1.2.4 ± Let us now state our main theorem. For this, recall that ifXis a smooth algebraic variety of pure dimensiondX, an algebraic mixedQ-Hodge moduleMonXis said to be smooth ifrat(M)[ d_X] is a local system, where

rat:D^b(MHM(X;Q))!D^bc(X;Q)

is the composition of real_X;Qand the forgetful functor in (1.2.3 1). The main result of this work is the following theorem:

1.2.5 ± THEOREM. Let X be a smooth projective algebraic variety and M;N be two complexes of mixed Hodge modules on X. IfMhas smooth cohomology i.e. the mixed Hodge moduleHⁱ(M)is smooth for every i2Z, then the morphism(1.2.2 1)is an isomorphism.

1.2.6 ± Let us denote by MHM(X;Q)_smthe full subcategory of MHM(X;Q) formed by the smooth mixedQ-Hodge modules and by D^b_sm(MHM(X;Q)) the full subcategory of D^b(MHM(X;Q)) formed by the complexes with smooth cohomology. Since MHM(X;Q)_sm is an abelian subcategory of MHM(X;Q) stable by extensions, the category D^b_sm(MHM(X;Q)) is a triangulated subcategory and 1.2.5 ensures it can be seen via (1.2.1 1) as a full triangulated subcategory of D_H^b (X;Q)_D.

1.2.7 ± In the proof of 1.2.5 we use essentially the assumption thatMis smooth. Since the definition of DH^b (X;Q)_D is really crude in comparison with the definition of the category of mixed Hodge modules, it may be that one has also to refine the construction of mixed Hodge complexes to get a positive answer to 1.2.2 in general.

1.3 ± Sketch of proof and content of the paper

1.3.1 ± The proof of 1.2.5 goes by reduction to the zero dimensional case.

As shown by M. Saito, a smooth mixed Hodge module is nothing but an admissible variation of mixed Hodge structures. Moreover since such a variation may also be seen as a mixed Hodge complex, to prove 1.2.5 we have to check that

Hom_D^b_(MHM(X;Q))(V^H;N )!Hom_D^b

H(X;Q)D(V^H;real(N ))

is an isomorphism for any admissible variation of mixed Hodge structures Vand anyN 2D^b(MHM(X;Q)). The idea of the proof is then very simple:

Step 1 We show that one may assume the variationVto be the trivial one Q^H_X. To do this reduction, we exploit the fact that the tensor category VMHS…X;Q†_ad of admissible variations of mixed Hodge structures is rigid. Namely we construct both on D^b(MHM(X;Q)) and DH^b (X;Q) a structure of module over VMHS…X;Q†_ad. The rigidity of the tensor cat- egory VMHS…X;Q†_adprovides then two isomorphisms

Hom_D^b

H(X;Q)D(V^H;M)ˆHom_D^b

H(X;Q)D(Q^H_X;V^_ M) (1:3:1 1)

Hom_D^b_(MHM(X;Q))(V^H;N )ˆHom_D^b_(MHM(X;Q))(Q^H_X;V^_ N ) (1:3:1 2)

whereV^_is the dual ofV. We check that the realization functor real is a morphism of modules and the compatibility of the isomorphisms (1.3.1 1) and (1.3.1 2) with real ensures that it is enough to consider the trivial variationQ^H_X . This step does not use the projectivity assumption and relies heavily on the notion of noncharacteristic mixed Hodge modules.

Step 2 Let p:X!Spec(C) be the structural morphism. Since p is as- sumed to be projective, we have a direct image functorp:D_H^b(X;Q)_D ! D_H^b(Spec(C);Q)_D for mixed Hodge complexes and we check that there is an isomorphism

Hom_D^b

H(X;Q)D(Q^H_X;M)ˆHom_D^b

H(SpecC;Q)(Q(0);pM) 8M 2DH^b (X;Q)_D which is compatible via real with the adjunction isomorphism in the derived categories of mixed Hodge modules provided by the adjoint functors p`p. We are then reduced to the case Spec(C) which was done in [20, 2.10] or in [1].

1.3.2 ± Let us now briefly run through the content of the paper. In the second and third sections we provide some complements onD-modules and mixed Hodge modules which are needed as preliminaries to the construc- tions involved in the proof. Though some of them may be well known to some reader they either do not appear in the literature or the proofs are too quickly sketched for our need. The proof of the main theorem is done in the fifth section while the fourth section contains recollections on mixed Hodge compleces and the last section the necessary though tedious constructions that are needed in the proof but may be skipped in a first reading.

1.3.3 ± In this paper a complex algebraic variety is a separated quasi- projective reduced scheme of finite type over SpecC. If X is a smooth complex algebraic variety, we denote by X^an the associated complex analytic manifold. Given an increasing filtrationW and an integern2Z, one denotes byWfngthe filtrationWfng_kˆW_{k n}.

2. Mixed Hodge modules

2.1 ± Side changing functors for filteredD-modules

2.1.1 ± LetXbe a smooth complex algebraic variety or a complex analytic manifold of pure dimensiondX. We denote byDX the ring of differential operators on X and by F its filtration by the order. Recall that D_X ˆ [_kF_kD_X,F₀D_X ˆO_X andF_kD_X ˆ0 for anyk<0. MoreoverD_X is a locally freeO_X-module andF_kD_X is a locally freeO_X-module of finite rank for anyk2Z. A filtered rightD_X-module is a pair (M;F) whereM is a right DX-module and FkM (k2Z) is an increasing sequence of additive subgroups ofMsuch thatF`MF<sub>k</sub>DX F<sub>k‡`</sub>M. We say that

Fis exhaustive if[_kF_kMˆM;

F is discrete if there exists locally an integer k₀2Z such that F_kMˆ0 fork<k₀.

One defines similarly the notion of filtered left D_X-modules. In the sequel all filtrations onDX-modules will be assumed to be exhaustive and discrete. The filtered right DX-modules form a quasi-abelian category MF(D_X) and we have a forgetful functorv:MF(D_X)!M(D_X).

2.1.2 ± REMARK. In [17] M. Saito uses rightD-modules almost exclu- sively, here it will be convenient to use also leftD-modules. Such a use will

be indicated by the symbol L e.g. MF(DX)^L will denote the category of filtered leftDX-modules.

2.1.3 ± Let v_X :ˆV^d_X^X be the invertible O_X-module of top differential forms onX. This is a rightD_X-module via the Lie derivative which may also be considered as a filtered rightD_X-module (v_X;F) for the filtrationF defined by Gr^F_kv_X ˆ0 ifk6ˆd_X. The category of filtered leftD_X-modules and filtered rightDX-modules are equivalent via the side changing functor which associates with a filtered leftDX-module (M;F) the filtered right D_X-module

(M;F)^r:ˆ(M;F)OX (v_X;F):

The underlying rightDX-module isM^rˆMOXvXand the filtration is given by F_kM^rˆF_k‡dXM_OXv_X. Let (v_X ¹;F) be the inverse of v_X with the filtrationFdefined by Gr^F_kv_X ¹ˆ0 ifk6ˆd_X, then given a filtered rightD_X-module (M;F)

(M;F)^`ˆ(v_X ¹;F)OX (M;F)

is the associated filtered left DX-module. We have FkM^` ˆ v_X ¹OX F_{k dX}M.

2.1.4 ± We now add the weight filtration to the picture. Let MF(D_X;W) be the following category. An object is a triple (M;F;W) where (M;F) is a filtered right D_X-module andW is a finite increasing filtration ofM by sub-DX-modules. A morphism is simply a morphism of filtered DX- modules which is compatible with the additional filtration W. The side changing filtered D_X-module (v_X;F) is then considered as an object (v_X;F;W) in MF(D_X;W) where W is the trivial filtration defined by Gr^W_n v_X ˆ0 if n6ˆd_X. The side changing equivalences 2.1.3 extend im- mediately to equivalences between the categories ofW-filtered filtered left DX-modules andW-filtered filtered rightDX-modules. The extension is given by

(M;F;W)^r:ˆ(M;F;W)OX(vX;F;W)

for a W-filtered filtered left DX-module (M;F;W). In other words the underlying filtered rightD_X-module is (M;F)^rand the filtrationW sat- isfiesW_kM^rˆ(W_{k dX}M)^r. Similarly if (M;F;W) is aW-filtered filtered rightD_X-module :

(M;F;W)^`ˆ(v_X ¹;F;W)_OX (M;F;W) (2:1:4 1)

is the associated W-filtered filtered left D-module. In (2.1.4 1) the fil- trationWonv_X ¹is defined by Gr^W_nv_X ¹ˆ0 ifn6ˆdX, the behaviour of the weight filtration is thus given byW_kM^{`</sup>ˆ(W<sub>k‡dX</sub>M)<sup>`}.

2.1.5 ± LetMbe a leftD_X-module and DR(M) be its de Rham complex. If (M;F) is a filtered left D_X-module, then DR(M) inherits a filtration defined forn50 andk2Zby

F_kDR(M)ⁿ ˆVⁿ_{X OX}F_k‡nM (2:1:5 1)

and the resulting filtered complex DR(M;F) is a complex of filtered differential operators of order 41. One sets ^pDR(M):ˆDR(M)[d_X].

Recall that the shift functor does not affect the Hodge filtration i.e FkpDR(M)ⁿˆFkDR(M)^n‡dX.

2.1.6 ± LetMbe a rightDX-module and Sp(M) be its Spencer complex.

If (M;F) is a filtered rightD_X-module, then Sp(M) inherits a filtration defined forn40 andk2Zby

F_kSp(M)ⁿ ˆF_k‡nM_OX^ⁿ U_X (2:1:6 1)

and the resulting filtered complex Sp(M;F) is a complex of filtered dif- ferential operators of order41.

2.1.7 ± IfMis a leftD-module then there is a functorial isomorphism c_M:Sp(M^r)! ^pDR(M);

if (M;F) is a filtered leftD-module then this morphism is compatible with the filtrations defined in (2.1.5 1), (2.1.6 1) and is therefore an iso- morphism of complexes of filtered differential operators of order41.

2.1.8 ± Let (M;F;W) be an object in MF(D_X;W). We denote by Sp(M;W) the Spencer complex of M endowed with the increasing filtration induced byW, we have then

Gr^W_n Sp(M;W)ˆSp(Gr^W_n M):

Assume that X is a smooth algebraic variety and (M;F)2MF(DX), (N ;F;W)2MF(D_X;W), then one defines

Sp_an(M;F)ˆSp(M^an) Sp_an(N ;F;W)ˆSp((N ;W)^an):

which are (filtered) complexes of sheaves ofC-vector spaces onX^an.

2.2 ± MixedQ-Hodge modules

2.2.1 ± Let X be a smooth algebraic variety of pure dimension dX and RˆQ;C. We denote by D^b(X;R) the bounded derived category of the abelian category of sheaves ofR-vector spaces onX^an. Inside D^b(X;R) one may consider the strictly full triangulated subcategory D^b_c(X;R) formed by the complexes K such that Hⁱ(K) is algebraically constructible for every i2Z. Let us denote by Perv(X;R) the heart of the perverset-structure on D^bc(X;R) and by^pHⁱthe perverse cohomology functors. By the Riemann- Hilbert correspondence, the de Rham and Spencer functors

are triangulated equivalences which are exact for the standardt-structure on the left hand-sides and the perverset-structure on the right hand-side.

In particular, it induces an exact equivalence of abelian categories between Mod_rh(D_X) and Perv(X;C).

2.2.2 ± REMARK. For convenience we will consider the category Perv(X;R)^L obtained by a shift in the definition of perversity: namelyK lies in Perv(X;R)^L if and only ifK[dX] belongs to Perv(X;R). This is for example the definition of perversity used in [16, 3.3-3]. With this convention aR-local systemVonXlies in Perv(X;R)^L, and ifMis in D^b_rh(D_X)^Lthen its de Rham and holomorphic solution complexes

DRan(M) RHomDXan(M^an;OX^an) lie also in Perv(X;C)^L.

2.2.3 ± Let us recall the definition of the basic category at the bottom of the notion of algebraicQ-Hodge modules introduced by M. Saito (see [17, 5.1.1]). Let MF_rh(D_X) be the category of filtered rightD_X-module (M;F) which satisfy the following two conditions:

(M;F) is a coherent filteredD_X-module, in other wordsFis a good filtration onM;

M is a regular holonomicD_X-module.

Since for an holonomic rightD_X-moduleM, the Spencer complex Sp_an(M)

is a perverse sheaf, we have a functor

MF_rh(D_X)!^v Mod_rh(D_X)ƒƒƒ!^Span Perv(X;C) and the following definition is meaningful:

2.2.4 ± DEFINITION. The category MFrh(DX;Q) is the fibre 2-product of the categories MF_rh(DX) and Perv(X;Q) over Perv(X;C):

MF_rh(D_X;Q):ˆMF_rh(D_X)_{Perv (X;C)}Perv(X;Q):

An object in MF_rh(D_X) is called a regular holonomic filteredD_X-module with aQ-structure.

Letn2Zbe an integer. Via induction on dimension of the support, the category MH(X;Q;n) of polarizable Hodge modules of weightnis defined as a strictly full subcategory of MF_rh(DX;Q). The precise definition of a Hodge module of weight nis given in [17, 5.1.6] while the polarizability condition is defined in [17, 5.2.10].

2.2.5 ± REMARK. Note that we do require pure Hodge modules to be polarizable, this is essential in many respects, the stability of mixed Hodge complexes by proper direct images being only one example where this condition is needed.

B y definition an object M in MF_rh(D_X;Q) is a 5-uplet K_Q;K_C;K_D;a^M_Q;a^M_D

where K_D :ˆ(M;F) is an object in MF_rh(D_X), forRˆQ;C,KRis an object in Perv(X;R) anda^M_Q;a^M_D are isomorphisms

K_QQCƒƒƒ!^a

M

Q K_C Sp_an(M)ƒƒ!^a

M D K_C:

A morphismu:M!M⁰is a triple (uQ;uC;uD) of morphisms that com- mute with the comparison morphisms.

2.2.6 ± REMARK. The category MF_rh(D_X;Q) is also equivalent to the category with objects triples M ˆ(K_Q;K_D;a_M) where K_D :ˆ(M;F) belongs to MF_rh(D_X), K_Q is an object in Perv(X;Q) and a_M : KQQC!Sp_an(M) is an isomorphism in Perv(X;C). The morphisms are simply pairs (uQ;uD) of morphisms such that

a_M⁰(u_QQC)ˆSp_an(u_D)a_M:

2.2.7 ± As in [17, 5.1.14] one defines MFrhW(DX;Q) to be the category of objects in MF_rh(D_X;Q) endowed with a finite increasing filtrationi.e.

MF_rhW(D_X;Q):ˆMF_rh(D_X;Q;W)Perv (X;C;W)Perv(X;Q;W)

where Perv(X;R;W) is the category of R-perverse sheaves with a finite increasing filtration. By definition the abelian category MHW(X;Q) is the strictly full subcategory of MF_rhW(D;Q) formed by the objectsM such that

Gr^W_n M2MH(X;Q;n);

and by its inductive construction the abelian category of algebraic mixed Hodge modules MHM(X;Q) is a strictly full subcategory of MHW(X;Q).

Note that the mixed Hodge modules are required to be polarizable.

2.2.8 ± When using leftDX-modules instead of rightDX-modules, it will be convenient to consider the category

MF_rh(D_X;Q)^L:ˆMF_rh(D_X)^L_{Perv (X;C)}LPerv(X;Q)^L

where the de Rham complex DRanof a leftD_X-module is used instead of the Spencer complex. One defines similarly the category MF_rhW(D_X;Q)^L. The side changing functors 2.1.3 , 2.1.4 and the isomorphisms 2.1.7 provide equivalences

MF_rh(D_X;Q)^L ƒƒ!^{( )r}

( )^`MF_rh(D_X;Q) MF_rhW(D_X;Q)^L ƒƒ!^{( )r}

( )^`MF_rhW(D_X;Q) such that forM 2MF_rhW(D_X;Q)^L

Gr^W_n M^r

ˆ Gr^W_{n dX}M_r :

We therefore define MH(X;Q;n)^Lto be the subcategory of MF_rh(D_X;Q)^L formed by the objectsM such thatM^rlies in MH(X;Q;n‡d_X) and the category of left mixed Hodge modules MHM(X;Q)^Las the subcategory of MFrhW(DX;Q)^Lformed by theMsuch thatM^rlies in MHM(X;Q).

2.3 ± Variations and mixed Hodge modules

2.3.1 ± LetX be a smooth algebraic variety andV a variation of mixed Hodge structures onX. As part of the definition, we have the following data:

a pair (V_Q;W_Q) whereV_Qis aQ-local system (²) onX^anandW_Qis a finite increasing filtration ofV_Qby local subsystems;

(²) IfAis a field aA-local system on a topological space is a locally constant sheaf of finite dimensionalA-vector spaces.

a triple ((V ;r);F ;W) where (V ;r) is a flat connexion on Xwith regular singularities at infinity,F is a finite decreasing filtration of V by locally freeO_X-module of finite rank which satisfies Griffiths' tranversality:

r:F ^kV !V¹_XOX F ^{k 1}V

andWis a finite increasing filtration ofV by locally freeO_X-modules of finite rank which are stable underr:

r:W_nV !V¹_XOX W_nV ; an isomorphism of filteredC-local systems

(VQ;WQ)QC ƒ!^aV (Kerr^an;W)

where the flat sections Kerr^anof the analytic connexion associated withrare endowed with the filtration induced byW.

All variations considered in this work are assumed to be polarizable (i.e.for a mixed variationV, all the pure variations Gr^W_n V are polarizable). The condition of admissibility was defined in [21, 10], the polarizability being part of it. The connexionrdefines onV a structure of leftD_X-module and setting F_kV :ˆF ^kV , Griffiths' transversality condition ensures that (V ;F) is a filtered leftD_X-module.

2.3.2 ± REMARK. For any variationVtheO_X-modules Gr^k_FGr^W_n V are locally free of finite rank and since V is an integrable connexion Char(V )ˆT_XX.

2.3.3 ± As shown by M. Saito (see [19, 3.27] and the remark after), the category VMHS…X;Q†_ad of admissible variations of Q-mixed Hodge structures is equivalent to the category of smooth mixed Hodge modules, more precisely the equivalence is given by

VMHS…X;Q†_ad!MHM(X;Q)_sm V 7!V^H

where V^H :ˆ(^HV)^r is the mixed Hodge module obtained by the side changing functor 2.1.4 from the left mixed Hodge module

HV:ˆ((VQ;WQ);(V ;F;W);a^H_V)

wherea_V^His the filtered isomorphism in the derived category given by the filtered quasi-isomorphism

aHV:(V_Q;W_Q)_QCƒ!^aV Ker(r^an;W),!DR_an(V ;W):

In particular if V is a pure variation of weight n then ^HV lies in MH(X;Q;n)^L or equivalentlyV^H belongs to MH(X;Q;n‡d_X).

3. Noncharacteristic mixed Hodge modules

3.1 ± FilteredD-modules and noncharacteristic inverse image

3.1.1 ± Letf :X!Ybe either a morphism of smooth complex algebraic varieties or a morphism of complex analytic manifolds. Let us say that a O_Y-moduleE isO_X-tor independent if

Tor_i^{f 1OY}(f ¹E;O_X)ˆ0 8i>0:

(3:1:1 1)

3.1.2 ± DEFINITION. A coherent filtered rightD_Y-module (M;F) is said to be noncharacteristic with respect tof if

M is noncharacteristic with respect tof;

for any k2Z and i>0 the O_Y-module Gr^F_kM is O_X-tor in- dependenti.e.

Tor_i^{f 1OY}(f ¹Gr^F_k M;OX)ˆ0:

3.1.3 ± REMARK. This is the condition introduced in [17, 3.5.1]. We may give a similar definition for filtered leftDY-modules. Being non character- istic in the sense of 3.1.2 is then a property stable under the side changing functors 2.1.3 i.e.a filtered rightD_Y-module (M;F) is noncharacteristic if and only if its associated filtered leftD_Y-module (M;F)^`is.

3.1.4 ± Letf :X!Ybe either a morphism of smooth complex algebraic varieties or a morphism of complex analytic manifolds. Letdˆd_X d_Ybe the relative dimension. As usual, we denote by (DX!Y;F) and (DY X;F) the filtered transfer modules (seee.g.[13, 5.1] for the filtered versions). If (M;F) is a filtered leftD_Y-module, one sets

f_O(M;F):ˆ(DX!Y;F)_f ¹_DYf ¹(M;F):

The underlyingOX-module isOX_f ¹_OYf ¹Mand the filtration is simply

given by

F_kf_O M:ˆIm O_Xf ¹_OY f ¹F_kM!f_OM :

The definition for filtered right D_Y-modules goes by the side changing functors, namely if (M;F) be filtered rightD_Y-module, then one sets (see [17, 3.5.1])

f_O(M;F):ˆf ¹(M;F)_f ¹_DY (D_{Y X};F):

3.1.5 ± If (M;F;W) is aW-filtered filtered left-D_X-module, one sets f_O(M;F;W):ˆ(DX!Y;F)_f ¹_DY f ¹(M;F;W):

The filtrationWis simply given by

W_k(f_OM):ˆImf_O(W_kM)!f_O M :

The definition for rightD_Y-modules goes by the side changing functors 2.1.4 .

3.2 ± Behaviour of weights in the noncharacteristic case

3.2.1 ± Let X be a smooth algebraic variety. The characteristic variety Char(M) of a mixed Hodge moduleMonXis the characteristic variety of the underlying rightD-module. It is a closed subvariety inTX. Though it is an abuse of notation, we will still denote by Char(M) the associated closed subset inTX^an.

3.2.2 ± LEMMA. LetM a mixed Hodge module on X. Then in TX^an Char(M)ˆSS(rat(M))

(3:2:2 1)

where SS denotes the microsupport defined by M. Kashiwara and P.

Schapira[12, 5.1.2].

PROOF. Let M be the left D_X-module associated with the under- lying rightD-module ofMandD(M) be its dual. SinceMis holonomic by [12, 11.3.3] and e.g.[6, 2.6.12]:

Char(M)ˆChar(M)ˆChar(DM)ˆSS(RHom_DXan((DM)^an;O_X^an))

ˆSS(RHomD_Xan(DM^an;O_X^an))

and the last term is isomorphic to rat(M)[ dX] in D^bc(X;Q) (see e.g.

[6, 4.2.1]). Since the microsupport does not change under shift [12, 5.1.3]

we have the equality (3.2.2 1) as desired. p

3.2.3 ± We say that a mixed Hodge moduleM2MHM(Y;Q) is nonchar- acteristic with respect to a morphismf :X!Yif and only if for any integer n2Zthe underlying filtered rightD_Y-module of Gr^W_n Mis noncharacter- istic with respect tof in the sense of 3.1.2.

3.2.4 ± PROPOSITION.Let f :X!Y be a morphism of smooth algebraic complexvarieties andM 2MHM(Y;Q). Assume thatMis noncharacter- istic with respect to f . Then we have an isomorphism

H^dfM '(H ^df^!M)( d) inMHM(X;Q).

PROOF. Let M₁ andM₂ be two objects in D^b(MHM(Y;Q)). The six operations formalism provides a natural morphism in D^b(MHM(X;Q))

f^!M₁fM₂!f^!(M₁M₂):

In particular takingM₁ˆQ^H_Y andM₂ˆM we get a morphism f^!Q^H_Y fM!f^!M

(3:2:4 1)

which gives the classical morphism (e.g. [12, 3.1.11])

f^!rat(Q^H_Y)frat(M)!f^!rat(M) (3:2:4 2)

onceratis applied. By 3.2.2 -rat(M) is noncharacteristic with respect tof in the sense of [12, 5.4.12] and therefore [12, 5.4.13] ensures that (3.2.4 2) is an isomorphism. Since rat is a conservative functor this implies that (3.2.4 1) is also an isomorphism. Since Y are smooth, we have an iso- morphism p^!_YQ^H 'Q^H_Y(d_Y)[2d_Y] and similarly forX, thereforef^!Q^H_Y is isomorphic to Q^H_X(d)[2d]. Hence (3.2.4 1) provides an isomorphism

fM(d)[2d]'f^!M

and it is enough to apply the cohomological functor H^d to get the

proposition. p

3.2.5 ± COROLLARY. Under the assumption of3.2.4 ,ifM is a pure Hodge module of weight n then H^dfM is pure Hodge module of weight n‡d.

PROOF. Indeed ifMany mixed Hodge module in MHM(Y;Q) then we know [19, 2.26] that

Gr^W_i (H^jfM)ˆ0 fori j>n (resp:Gr^W_i (H^jf^!M)ˆ0 fori j<n) if Gr^W_i Mˆ0 for i>n (resp. i<n). Hence the purity assumption im- plies (³) on the one hand that

Gr^W_i (H^dfM)ˆ0 fori>n‡d and on the other hand that

Gr^W_i ((H ^df^!M)( d))ˆ0 fori n‡d:

The corollary follows then from 3.2.4 . p

3.3 ± A remark on Cauchy-Kovalevskaya-Kashiwara theorem

3.3.1 ± Let f :X!Y be a morphism of complex manifolds. We let d:d_X d_Y be the relative dimension. LetM;N in D^b_coh(D_Y). Consider the canonical morphism in D^b(X;C)

can(f;M;N ):f ¹RHomDY(M;N )!RHomDX(Lf_OM;Lf_O N ) (see e.g. [6, p. 106] for its definition), which provides in particular the fol- lowing morphisms

can(f;M;OY):f ¹RHomDY(M;OY)!RHomDX(Lf_OM;OX) (3:3:1 1)

can(f;OY;M):f ¹RHomDY(OY;M)!RHomDX(OX;Lf_OM):

(3:3:1 2)

Assume thatMis noncharacteristic with respect tof. Then (3.3.1 1) is an isomorphism. This result is the generalization of the Cauchy-Kovalevskaya theorem proven by M. Kashiwara in [11] (see also e.g. [6, Theorem 4.3.2]).

Using duality, it follows that (3.3.1 2) is also an isomorphism. However this is a morphism in the derived category. In the sequel, see Section 6, we rather need a genuine morphism of complexes that represents it. We pro- vide now its definition.

3.3.2 ± For this let us remark that, given a leftDY-moduleM, there exists a functorial morphism of complexes ofCX-modules

f_M^[ :f ¹DR(M)!DR(f_OM) (3:3:2 1)

(³) The Tate twist is defined in [19, 2.17.7].

which forM ˆOY coincides with the canonical morphism f ¹V_Y!V_X. Then-th component of (3.3.2 1) is the morphism of sheaves

(f_M^[ )ⁿ:f ¹Vⁿ_Yf ¹_OY f ¹M !Vⁿ_XOX f_OM defined for a local sectionv2Vⁿ_Y and a local sectionm2Mby

(f_M^[ )ⁿ(vm)ˆfv(1m):

3.3.3 ± REMARK. This formula defines a morphism of complexes. Indeed by the Leibniz formula it is enough to consider the case wherevis a function inOY and we may further assumevˆ1. We may further assume that we have a coordinate system (y1;. . .;ys) on Y and a coordinate system (x1;. . .;xr) onX. We have then for a local sectionm2M

f_M^[ …r_M(1m)† ˆf_M^[ X^s

jˆ1

dy_j@_yjm

!

ˆX^s

jˆ1

fdy_j(1@_yjm)

ˆX^s

jˆ1

X^r

iˆ1

@xifjdxi(1@yjm)ˆX^r

iˆ1

dxi X^s

jˆ1

@xifj@yjm

!

ˆX^r

iˆ1

dx_i X^s

jˆ1

@_xif_j@_yjm

!

ˆX^r

iˆ1

dx_i@_xi(1m)

ˆ rf_OM(f_M^[ (1m)):

3.3.4 ± Recall that given a left DX-module M, one has a canonical iso- morphism of complexes DR(M)ˆHomDX(Sp(DX);M). Since Sp(DX) is a resolution of O_X by locally freeD_X-module, this isomorphism provides a natural isomorphism

co(M):DR(M)!RHom_DX(O_X;M) in D^b(X;C):

3.3.5 ± LEMMA. Let M be a coherent left D_Y-module. If M is non- characteristic with respect to f , then the morphism(3.3.2 1)represents the morphism(⁴)

can(f;O_Y;M):f ¹RHom_DY(O_Y;M)!RHom_DX(O_X;Lf_OM):

In particular(3.3.2 1)is a quasi-isomorphism.

(⁴) Recall that ifM is noncharacteristic coherentDY-module, thenLf_OM is concentrated in degree 0 and thusLf_OMˆf_OM.

3.3.6 ± REMARK. The filtered inverse image of a differential complex of order41 is given by the tensor product

f_Diff (M;F):ˆ(V_X;F)_f 1V_Y f ¹(M;F):

In particular the filtration on the inverse image is given by Fk(f_Diff M)ⁿˆ X

p‡qˆn

Im(V^p_Xf ¹_OY f ¹Fk‡pM^q)

and the canonical morphism of DG-algebrasf ¹V_Y!V_Xinduces therefore a morphism

f_M^[ :f ¹(M;F)!f_Diff M:

Another way to interpret the morphism (3.3.2 1) is to remark that one has a canonical isomorphism of complexes

f_Diff DR(M):ˆV_Xf ¹_VY f ¹DR(M)! DR(f_O M):

With this description of the de Rham complex of the inverse image, (3.3.2 1) is then nothing but the morphism induced by the usual morphism f ¹V_Y !V_X.

3.3.7 ± REMARK. The morphism (3.3.2 1) is functorial inM, but it is also compatible with the composition of morphisms. More precisely, if

W!^g X!^f Y

are morphisms of smooth complex algebraic varieties, then (fg)^[_Mˆg^[_f

OMg ¹f_M^[ for any leftD_Y-moduleM.

IfMis a rightD_Y-module, using the isomorphism given in 2.1.7 , we get a morphism of complexes ofC_X-modules

f_M^[ :f ¹Sp_X(M)[d]!Sp(f_OM) (3:3:7 1)

which is functorial in M. Moreover 3.3.7 translates into the following equality

(fg)^[_M ˆg^[_f

OM(g ¹f_M^[ [e]) (3:3:7 2)

for a sequence of morphismsW!^g X!^f Yof smooth algebraic varieties. In (3.3.7 2) the integer e is the relative dimension eˆd_W d_X of the morphismg.

3.4 ± Noncharacteristic inverse image

3.4.1 ± Let Mˆ((M;F);K_Q;a_M) be an object in MF_rh(D_Y;Q) and f :X!Y a morphism of smooth complex algebraic varieties of relative dimensiond. Assume thatMis noncharacteristic with respect tof.

3.4.2 ± REMARK. By 3.2.2 the perverse sheafKQis noncharacteristic with respect tof, in the sense of Kashiwara-Schapira. Hence

f ¹KQ[d]ˆf^!KQ[ d]

is a perverse sheaf onX^an. Moreoverf_O(M;F) is a regular holonomic fil- tered rightD_X-modulei.e.belongs to MF_rh(D_X).

The morphism (3.3.7 1) defined at the level of the complexes provides an isomorphismaf_OMin Perv(X;C):

Therefore, we have a well defined object

f_OM:ˆ(f_O(M;F);f ¹K_Q[d];a_fOM)

in MF_rh(D_X;Q). The construction is functorial for noncharacteristic ob- jects in MFrh(DY;Q).

3.4.3 ± PROPOSITION. Let M 2MH(Y;Q;n) be a Hodge module of weight n and f :X!Y a morphism of smooth complexalgebraic varieties. Assume that M is noncharacteristic for f . Then, its nonchar- acteristic inverse image f_OM is a Hodge module on X of weight n‡d.

3.4.4 ± Let us start now the recollections on vanishing cycles which are necessary to write down the proof of 3.4.3 . In order to rely more easily on [15] we will use the convention inloc.cit., in particular we will use left D-modules and left perverse sheaves as in 2.2.2 . Let X be a smooth algebraic complex variety andHbe asmooth hypersurface inX defined by a global equation g2G(X;OX). Denote by i:H,!X the closed

immersion. Consider the following diagram

where Ce denote the universal cover of the punctured complex plane C:ˆCn f0g. Let RˆQ;CandK2D^b_c(X;R), one has a distinguished triangle in D^bc(H;R)

i ¹K!Cg(K)!Fg(K)ƒ!^‡1

whereC_g(K):ˆi ¹Rpp ¹Kis the nearby cycles complex andF_g(K) is the vanishing cycles complex. IfK is aleft perverse sheaf onX^an, then both C_g(K) andF_g(K) areleft perverse sheavesonH^an.

3.4.5 ± LetMbe a left algebraicDX-module. Assume thatM is regular holonomic. Then by [15, 4.4-2]Mis specializable alongHand we denote by Vthe canonicalV-filtration ofMdefined by the relative order with respect toH. The nearby and vanishing cyclesD_H-modules ofMare defined as

C_g(M):ˆ M

14a<0

Gr^V_a(M)j_H; Fg(M):ˆ M

1<a40

Gr^V_a(M)j_H:

They are regular holonomic DH-modules [15, 4.7-5] and one has an iso- morphism (functorial inM) [15, 5.3-2] ofleft perverse sheavesonH^an

can^C_g;M:C_g(DR_X^an(M^an))!DR_H^an(C_g(M)^an) (3:4:5 1)

and a similar isomorphism can^F_g;Mfor the vanishing cycles. If (M;F) is a regular holonomic filtered leftD_X-module which isquasi-unipotent and regular along Hin the sense of [17, 3.2.1], then the filtered nearby and vanishing cycles functors are endowed with the filtration induced (⁵) byF and belong to MF_rh(D_H)^L.

(⁵) For filtered rightDX-module, the vanishing cycles are defined similarly in [17, 5.1.3.3] but endowed by the filtration induced byFf1g. The definition for filtered leftD-modules considered here is compatible with M. Saito's definition via the side-changing functors 2.1.3 .

3.4.6 ± LetM2MFrh(DX;Q)^Lsuch that the underlying filtered leftDX- module is quasi-unipotent and regular along H, then the isomorphism (3.4.5 1) provide an object in MF_rh(D_H;Q)^L

C_g(M):ˆ(C_g(M;F);C_g(K_Q);a^C(M)) a^C(M) :ˆC(a^M)(can^C_g;M) ¹ and one defines similarly F_g(M). By [15, 5.3-2] we have a commutative diagram in the category Perv(H^an;C)^L

such that the vertical sequences are short exact sequences. The morphisms iM!Li_OM and sp_g;M are given (functorially in M) in the derived category by the morphisms of complexes [15, 4.4-4,4.6-4]

where the middle term Gr^V₁Mj_H lies in degree 0. If (M;F) is non- characteristic with respect toi, then (M;F) is quasi-unipotent and regular alongH[17, 3.5.6] and we have Gr^V_a(M;F)ˆ0 as soon asa2f= 1; 2;. . .g.

This implies thatFgMˆ0 and thatNˆ0 onCg(M). We have therefore an isomorphism MFrh(DH;Q)^L

i_OM 'Cg(M) (3:4:6 1)

and the nilpotent monodromy operatorN onCg(M) vanishes. Using the side changing functors we get a similar result for rightD-modules.

3.4.7 ± Assume now thatMis a pure Hodge module of weightnonX(we use now the convention in [17] and thus in particular rightD-modules).

Let

g:Xƒ!^ig XA¹ƒ!^p A¹

be the factorization ofgby its graph. By [17, 3.2.4,3.2.5] we have a canonical isomorphism

c_g(M)'iC_g(M) (3:4:7 1)

wherec_g(M):ˆCp(igM) is the nearby cycles functor as defined by M.

Saito [17, 5.1.3.3]. SinceNis trivial onc_g(M), the associated monodromy filtration M is trivial and therefore the inductive construction of the cate- gory of pure Hodge modules [17, 5.1.6.2] implies that

Gr^Mfn_i ^1gc_g(M)ˆGr^M_{i n‡1}c_g(M)ˆ c_g(M) if iˆn 1

0 otherwise

is a pure Hodge module of weightionX. Then, it follows from (3.4.7 1), (3.4.6 1) and [17, 5.1.9] thati_OMis a pure Hodge module of weightn 1 onHas desired.

PROOF OF3.4.3 . Let

be the factorization of f via its graph. By [17, 3.5.4], it is enough to check the result forpandif. Forpthe result is a consequence of [19, 3.21] (see also [19, 3.28]) since one has an isomorphism

p_O MˆQ^H_X[dX]PM

andQ^H_X[dX] is a pure Hodge module of weightdX. Let us now prove the result for any closed immersioni:X,!Yby induction on the codimension

ofXinY. Since locally onXwe may factorias a composition of immersions of codimension one between smooth closed subschemes, by [17, 3.5.4] and the fact that the conditions which define Hodge modules are of local nature [17, 5.1.7], we are reduced to the case of a smooth closed hypersurface which

was considered above. p

3.4.8 ± Let Mˆ((M;F;W);(KQ;WQ);aM) be an object in MF_rhW(D_Y;Q) andf :X!Ya morphism of smooth complex algebraic varieties of relative dimension d. Assume that M is noncharacteristic with respect to f. The morphism (3.3.7 1) defined at the level of the complexes provides an isomorphismaf_OMin Perv(X;C):

where (K;W)fdg:ˆ(K;Wfdg)[d] and Wfdg_k :ˆW_{k d}. Therefore, we have a well defined object

f_OM :ˆ(f_O(M;F;W);f ¹(K_Q;W_Q)fdg;a_fOM)

in MF_rhW(D_X;Q). As before, the construction is functorial for non- characteristic objects in MF_rhW(D_Y;Q). A proof similar to the one of 3.4.3 yields the following:

3.4.9 ± PROPOSITION. LetM 2MHM(Y;Q)be a mixed Hodge module and f :X!Y a morphism of smooth complexalgebraic varieties. Assume that M is noncharacteristic for f . Then, its noncharacteristic inverse image f_OMis a mixed Hodge module on X.

3.4.10 ± REMARK. If M is noncharacteristic then we have an iso- morphism in the derived category D^b(MHM(X;Q))

f_OM 'fM[d]

wherefM is the pullback functor constructed in [19, 4.4].

4. Mixed HodgeD-complexes 4.1 ± Categories of diagrams

4.1.1 ± IfR is one of the symbolsQ;Cor D we letA_R be an additive category. We assume to be given on the homotopy category K^b(A_R) a null system N_R and one denotes by Q_R the associated multiplicative system.

4.1.2 ± We also assume to be given a full DG-subcategoryCRof (AR). As usual one denotes by

C_RˆZ⁰C_R K_R:ˆH⁰C_R

the associated category and the homotopy category, which are respectively full subcategories of C^b(AR) and K^b(AR). We assume that the objects ofCR

enjoy the following stability conditions:

ifa:A!A⁰is a morphism in C^b(A_R) which belongs toQ_R, thenA belongs toCRif and only ifA⁰belongs toCR;

ifa:A!A⁰ is a morphism in CR then its mapping cone is also an object inC_R, in particular ifA2C_RthenA[1]2C_R.

These conditions ensure that K_Ris a strictly full triangulated subcategory of K^b(A_R) and that D_R:ˆK_R[Q_R¹] is also strictly full triangulated sub- category of K^b(A_R)[Q_R¹] We also assume given two DG-functors

FR:CR!CC RˆQ;D

such that F_R(Q_R) Q_C and F_R(Mc(a))ˆMc(F_R(a)) for a2[A;A⁰]. In particular we have induced functors

FR:KR!KC FR:DR!DC

which are triangulated.

4.1.3 ± DEFINITION. A diagram of the form

(4:1:3 1)

consists then of the following data:

forRˆQ;C;D an objectA_Rin C_R;

two morphism in CC

a^A_R:FR(AR)!AC RˆQ;D which belongs toQ_C.

4.1.4 ± The complex of morphisms between two diagrams A and A⁰ is defined by

[[A;A⁰]]ⁿˆ aˆ(aQ;aC;aD;hQ;hD):

aQ 2[[AQ;A⁰_Q]]ⁿ aC 2[[AC;A⁰_C]]ⁿ a_D 2[[A_D;A⁰_D]]ⁿ hQ 2[[FQ(AQ);A⁰_C]]^{n 1} h_D 2[[F_D(A_D);A⁰_C]]^{n 1} 8>

>>

>>

>>

><

>>

>>

>>

>>

:

9>

>>

>>

>>

>=

>>

>>

>>

>>

; where the differential is given by

d a_Q a_C a_D hQ

hD

0 BB BB

@ 1 CC CC Aˆ

daQ

daC

daD

a_Ca^A_Q a^A_Q⁰F_Q(a_Q) dh_Q aCa^A_D a^A_D⁰FD(aD) dhD

0 BB BB BB B@

1 CC CC CC CA :

We get a DG-categoryC_D;Qand we denote by

C_D;QˆZ⁰C_D;Q K_D;QˆH⁰C_D;Q

the associated categories. A morphismaˆ(a_Q;a_C;a_D;h_Q;h_C) in C_D;Qis thus a 5-uplet such that u_R is a morphism of complexes andh_Q;h_D are graded morphisms of degree 1 whose differentials are equal to the defect in commutativity of the squares

4.1.5 ± By the definition of the mapping cone, the forgetful functors