Weight filtration of the limit mixed Hodge structure at infinity for tame polynomials
aa r X i v : . [ m a t h . AG ] O c t WEIGHT FILTRATION OF THE LIMIT MIXED HODGE STRUCTUREAT INFINITY FOR TAME POLYNOMIALS
ALEXANDRU DIMCA AND MORIHIKO SAITO
Abstract.
We give three new proofs of a theorem of C. Sabbah asserting that the weightfiltration of the limit mixed Hodge structure at infinity of cohomologically tame polynomialscoincides with the monodromy filtration up to a certain shift depending on the unipotentor non-unipotent monodromy part.
Introduction
Let X = C n ( n ≥ S = C . Let f : X → S be a cohomologically tame polynomialmap in the sense of [Sab3], i.e. there is a relative compactification f : X → S of f suchthat f is proper and the support of ϕ f − c R j ∗ Q X is contained in the affine space X (hencediscrete) for any c ∈ C , where j : X ֒ → X denotes the inclusion.Set m = n −
1, and X s := f − ( s ) ⊂ X for s ∈ S . There is a non-empty Zariski-opensubset S ′ of S such that X s is smooth for s ∈ S and the H i ( X s , Q ) form a local system on S ′ . For s ∈ S ′ , we have moreover H i ( X s , Q ) = 0 if i = 0 , m, H ic ( X s , Q ) = 0 if i = m, m, and H ( X s , Q ) = Q , H mc ( X s , Q ) = Q ( − m ). These follow from the discreteness of thesupport of ϕ f − c R j ∗ Q X together with X = C n by using the Leray spectral sequence as inRemark (1.2) below. For s ∈ S ′ , set H s := H m ( X s , Q ) , H cs := H mc ( X s , Q ) . By [De1], these have the canonical mixed Hodge structures which are dual of each other (upto a Tate twist). By the definition of the weight filtration W in loc. cit., Th. 3.2.5 (iii), wehave Gr Wm + k H s = Gr Wm − k H s = 0 for k <
0. In [DS], Th. 0.3, the following was shown:(0 . { Gr Wm + k H s } s ∈ S ′ , { Gr Wm − k H cs } s ∈ S ′ are constant on S ′ if k > { H s /W m H s } s ∈ S ′ and { W m H cs } s ∈ S ′ are constant.This is closely related to (1.1.3) below.Let H ∞ be the limit mixed Hodge structure of H s for s → ∞ , and similarly for H c ∞ , see[St1], [St2], [StZ]. Set N := (2 πi ) − log T u with T u the unipotent part of the monodromy atinfinity. This is an endomorphism of type ( − , −
1) of H ∞ , H c ∞ . Let L be the filtration on H ∞ , H c ∞ induced by the weight filtration W respectively on H s , H cs for s ∈ S ′ . Then theweight filtration W on H ∞ , H c ∞ coincides with the relative monodromy filtration of ( L, N ),see [StZ]. In particular, W on Gr Lm H ∞ , Gr Lm H c ∞ coincides with the monodromy filtrationshifted by m (i.e. with center m ).Let H ∞ , , H c ∞ , respectively denote the unipotent monodromy part of H ∞ , H c ∞ , andsimilarly for the non-unipotent part H ∞ , =1 , H c ∞ , =1 . By (0.1) we have(0 . H ∞ , =1 = Gr Lm H ∞ , =1 , H c ∞ , =1 = Gr Lm H c ∞ , =1 . We thus get the following well-known assertion (see also Appendix of [MT]):
Proposition 1.
With the above notation and assumption, the weight filtration W on H ∞ , =1 , H c ∞ , =1 coincides with the monodromy filtration shifted by m . In this paper we give three new proofs of the following.
Theorem 1 (C. Sabbah).
With the above notation and assumption, the weight filtration W on H ∞ , , H c ∞ , coincides with the monodromy filtration shifted by m +1 and m − respectively,and we have the isomorphisms of mixed Hodge structures for k ≥ Wm + k H s ∼ = Gr Wm + k Coker( N | H ∞ , ) , Gr Wm − k H cs ∼ = Gr Wm − k Ker( N | H c ∞ , ) , where Coker( N | H ∞ , ) is a quotient of H ∞ , ( − and Ker( N | H c ∞ , ) ⊂ H c ∞ , . Note that the assertions on H ∞ , and H c ∞ , are dual of each other. The last assertion ofTheorem 1 means that the primitive part of the graded pieces of the monodromy filtrationon H ∞ , is given by Gr Wm + k H s for k ≥
1, and the coprimitive part for H c ∞ , by Gr Wm − k H cs .Theorem 1 was first obtained by C. Sabbah as a corollary of [Sab3], Th. 13.1 where heuses a theory of Fourier transformation, Brieskorn lattices, and spectra at infinity, which wasdeveloped by him (see also [Sab1], [Sab2]). Recently another proof has been given also byhim in Appendix of [MT] without using Brieskorn lattices or spectra at infinity, but usingFourier transformation where irregular D -modules inevitably appear.It does not seem, however, that the above theory is absolutely indispensable for the proofof Theorem 1. In fact, the theorem was almost proved in [DS] where the following was shown(see also [Di1], 4.3–5): Theorem 2 ([DS], Th. 0.3).
With the above notation and assumption, let ν k and ν ′ k denotethe number of Jordan blocks of size k for the monodromy on H ∞ , and Gr Lm H ∞ , respectively.Let s ∈ S ′ . Then ν k = dim Gr Wm + k H s , ν ′ k = ν k +1 for any k ≥ . We give the first proof of Theorem 1 in this paper by showing that Theorem 2 impliesTheorem 1 using some lemma of linear algebra, see (1.3–4) below.The second proof of Theorem 1 in this paper uses a geometric argument together withduality, and is quite different from (and perhaps more intuitive than) the one in the proofof Th. 0.3 in [DS]. It is finally reduced to the following:
Proposition 2.
Let ι s : H cs → H s be the natural morphism of mixed Hodge structures for s ∈ S ′ . Then it induces an isomorphism of Hodge structures Gr Wm ι s : Gr Wm H cs ∼ −→ Gr Wm H s for s ∈ S ′ . We give two proofs of Proposition 2 in this paper. One proof uses semisimplicity of pureHodge modules together with a certain property of the mixed Hodge module H f ∗ ( Q h,X [ n ])coming from the cohomologically tame condition. (For Q h,X , see (1.1) below.) Anotherproof uses positivity of the polarization on the primitive cohomology of a compact K¨ahlermanifold together with Hironaka’s resolution of singularities.The third proof of Theorem 1 in this paper is given as a corollary of Theorem 3 below,which holds for any pure Hodge module M of weight n on S without a constant directfactor, and was proved by C. Sabbah in Appendix of [MT]. For a bounded complex of mixedHodge modules M • on a complex algebraic variety X in general, we denote the mixed EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 3
Hodge structure H j ( a X ) ∗ M • by H j ( X, M • ) (using a remark before (1.1.5) below), where a X : X → pt is the canonical morphism. This notation is compatible with that of thecohomology of the underlying Q -complex. Theorem 3 (C. Sabbah).
Let M be a pure Hodge module of weight n on S having noconstant direct factor, i.e. H − ( S, M ) = 0 . Set H := ψ /t, M . Then the weight filtration on H coincides with the monodromy filtration shifted by n , and the N -primitive part P Gr Wk H is given by Gr Wk H ( S, M ) for any k where they vanish unless k ≥ n . The proof of Theorem 3 in this paper uses the notion of representative functor and theuniversal extension f M of a pure Hodge module M by a constant mixed Hodge module (see(3.1) below), but Fourier transformation is not used. Note that f M was defined in loc. cit.by using a sheaf-theoretic operation explicitly. By the property of the universal extensionwe have f M / M = a ∗ X H ( S, M )[1] , H − ( S, f M / M ) = H ( S, M ) . The proof of Theorem 3 is reduced to the comparison between the global universal extensionon S and the local one on a neighborhood of ∞ ∈ P for the underlying perverse sheaves,where some argument is similar to the one in the proof of [DS], Th. 0.3.We thank the referee for helping us to improve the paper.The first named author was partially supported by the grant ANR-08-BLAN-0317-02(SEDIGA). The second named author is partially supported by Kakenhi 21540037.In Section 1 we explain some basics on cohomologically tame polynomials, and give theproof of Theorem 1 using Theorem 2 after showing Lemma (1.3). In Section 2 we give twoproofs of Proposition 2, and then a geometric proof of Theorem 1 after showing Lemma (2.3).In Section 3 we explain the notion of a universal extension by a constant sheaf, and thenprove Theorem 3 which implies Theorem 1.
1. Cohomologically tame polynomials
In this section we explain some basics on cohomologically tame polynomials, and give theproof of Theorem 1 using Theorem 2 after showing Lemma (1.3).
Set X = C n ( n ≥ S = C . Let f : X → S be a polynomial map, and f : X → S be an algebraiccompactification of f (i.e. f is proper). Let j : X ֒ → X be the inclusion. Note that R j ∗ Q X [ n ]is a perverse sheaf since j is an affine open immersion, see [BBD]. The intersection complexIC X Q is a subobject of the perverse sheaf R j ∗ Q X [ n ] (see loc. cit.) and the vanishing cyclefunctor ϕ f − c (see [De2]) is an exact functor of perverse sheaves (up to a shift). So we getthe first inclusion of(1 . .
1) supp ϕ f − c IC X Q ⊂ supp ϕ f − c R j ∗ Q X [ n ] = supp ϕ f − c R j ! Q X [ n ] . For the last isomorphism, we have the relation D ◦ R j ∗ = R j ! ◦ D and the compatibilityof ϕ f − c with the dualizing functor D , i.e. D ◦ ϕ = ϕ ◦ D where a Tate twist may appeardepending on the eigenvalue of the monodromy, see e.g. [Sai1], 5.2.3. ALEXANDRU DIMCA AND MORIHIKO SAITO
Assume now that supp ϕ f − c R j ∗ Q X is contained in the affine space X (hence discrete) forany c ∈ C . This means that f is a cohomologically tame polynomial in the sense of [Sab3].Consider the canonical morphisms(1 . . R j ! Q X [ n ] → R j ∗ Q X [ n ] , R j ! Q X [ n ] → IC X Q , IC X Q → R j ∗ Q X [ n ] . Since the restrictions of these morphisms to X are the identity morphisms, and ϕ commuteswith the direct images by proper morphisms, we get the following.(1.1.3) The direct images by f of the mapping cones of the canonical morphisms in (1.1.2)are isomorphic to direct sums of shifted constant sheaves on S .Indeed, the above properties imply constancy of the cohomology sheaves. Then (1.1.3)follows from the vanishing of Ext i ( Q S , Q S ) for i > τ ≤ k in[De1].Let Q h,X denote the object in D b MHM( X ) (the bounded derived category of mixed Hodgemodules on X ) which is uniquely characterized by the following two conditions: Its under-lying Q -complex is Q X , and its 0-th cohomology H ( X, Q h,X ) := H ( a X ) ∗ Q j,X has weight0, see [Sai2], 4.4.2. We have(1.1.4) Replacing Q X with Q h,X in (1.1.2), the assertion (1.1.3) holds in D b MHM( S ).Indeed, any admissible variation of mixed Hodge structure M on S is a constant variation,see e.g. [StZ], Prop. 4.19. (This follows from the existence of the canonical mixed Hodgestructure on H ( S, M ) by using the restriction morphisms H ( S, M ) → M s which aremorphisms of mixed Hodge structures for s ∈ S , where M s is the pull-back of M by theinclusion { s } ֒ → S , and M [1] is a mixed Hodge module on S .) So we get M = a ∗ S H for H ∈ MHS (the category of graded-polarizable mixed Hodge Q -structures in [De1]), where a S : S → pt is the canonical morphism. Note that MHM( pt ) is naturally identified withMHS, see [Sai2]. We have moreover for i > . .
5) Ext i MHM( S ) ( a ∗ S H, a ∗ S H ′ ) = Ext i MHS ( H, ( a S ) ∗ a ∗ S H ′ ) = Ext i MHS ( H, H ′ ) = 0 , since Ext i = 0 ( i >
1) in MHS by a well-known corollary of a theorem of Carlson [Ca](which implies the right-exactness of the functor Ext ( Q , ∗ )). So (1.1.4) follows by usingthe canonical filtration τ ≤ k in [De1]. Let f : X → S be as in the beginning of (1.1). We have the Leray spectralsequence in MHS:(1 . . E i,j = H i ( S, H j f ∗ ( Q h,X [ n ])) ⇒ H i + j + n ( X, Q ) , using the canonical filtration τ ≤ k as in [De1]. (This will be used later.)We have E i,j = 0 for i / ∈ [ − ,
0] since S = C . So (1.2.1) degenerates at E , and we get(1 . . H i ( S, H j f ∗ ( Q h,X [ n ])) = 0 for ( i, j ) = ( − , − n ) , since X = C n and H j f ∗ ( Q h,X [ n ]) = 0 for j ≤ − n (using the classical t -structure).Assume f is cohomologically tame (using an appropriate compactification of f as in (1.1)).Then(1 . . H j f ∗ ( Q h,X [ n ]) is constant for j = 0 , EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 5 by using the exactness of ϕ (up to a shift) together with the commutativity of ϕ and thedirect image under proper morphisms as in (1.1). So (1.2.2) implies(1 . . H j f ∗ ( Q h,X [ n ]) = 0 unless j = 1 − n or 0, Let V • be a finite dimensional graded Q -vector space with an action of N of degree − , i.e. N ( V k ) ⊂ V k − . Let V ′ • be a graded vector subspace stable by N . Set V ′′ • := V • /V ′ • . Let m be an integer. Assume the action of N on V ′′ • vanishes, and N k : V ′ m + k ∼ −→ V ′ m − k for any k ≥ . Set C ′ k := Coker( N : V ′ k +2 → V ′ k ) so that N induces δ k : V ′′ m + k +2 → C ′ m + k . Let ν k be thenumber of Jordan blocks of size k for the action of N on V • . Then ν k +1 = ( dim Coker δ + P j dim Ker δ j if k = 0 , dim Coker δ k + dim Im δ k − if k ≥ . Proof.
Let V ′ m + k be the primitive part defined by Ker N k +1 ⊂ V ′ m + k for k ≥
0. We have theprimitive decomposition(1 . . V ′ • = L k ≥ (cid:0)L kj =1 N j V ′ m + k (cid:1) with V ′ m + k ∼ −→ C ′ m + k . Set n k = dim Im δ k . For each k ≥
0, there are bases { v ′ k,j } j of V ′ m + k (= C ′ m + k ) and { v ′′ k,j } j of V ′′ m + k +2 togetherwith lifts v k,j of v ′′ k,j in V m + k +2 such that(1 . . N v k,j = ( v ′ k,j if 1 ≤ j ≤ n k , . Indeed, by the definition of δ k , the assertion is trivial if we consider the equality modulo N V ′ • , i.e. if we add the term + N u k,j for some u k,j ∈ V ′ m + k +2 on the right-hand side of (1.3.2).Then we can replace the lift v k,j of v ′′ k,j with v k,j − u k,j , and (1.3.2) is proved.The assertion of Lemma (1.3) now follows from (1.3.1) and (1.3.2). Assume the morphisms δ k : V ′′ m + k +2 → C ′ m + k in Lemma (1.3) are bijectivefor any k ≥
0. Then (1.3.2) in the proof of Lemma (1.3) implies that the the primitivedecomposition of V ′ m + • can be lifted to that of V m +1+ • . We show the assertion for H ∞ , sincethis implies the assertion for H c ∞ , by duality. We can replace H ∞ , with the graded piecesGr W • H ∞ , in order to define ν k , ν ′ k , since W is strictly compatible with N k for any k ≥ V k = Gr Wk H ∞ , , V ′ k = Gr Wk Gr Lm H ∞ , , V ′′ k = ( Gr Lk H ∞ , if k > m, k ≤ m. Here Gr Wj Gr Lk H ∞ , = 0 for j = k if k > m , since N = 0 on Gr Lk H ∞ , for k > m by (0.1).Using the primitive decomposition (1.3.1), we get(1 . . ν ′ k +1 = dim C ′ m + k for k ≥ . ALEXANDRU DIMCA AND MORIHIKO SAITO
We show that Theorem 2 together with Lemma (1.3) imply the isomorphism(1 . . δ k : V ′′ m + k +2 ∼ −→ C ′ m + k for k ≥ . By (1.5.1) the surjectivity of δ k is equivalent to(1 . . ν ′ k +1 = dim Im δ k for k ≥ , and we have by Theorem 2 and Lemma (1.3) ν ′ k +1 = ν k +2 = dim Coker δ k +1 + dim Im δ k for k ≥ . So (1.5.3) follows by decreasing induction on k ≥ δ k , we get by Lemma (1.3) together with the surjectivity of δ ν = dim V ′′ m +1 + P k ≥ dim Ker δ k , since δ − vanishes. We have moreover ν = dim V ′′ m +1 by Theorem 2. So the injectivity of δ k ( k ≥
0) follows. Thus (1.5.2) is proved.Then the primitive decomposition of V ′ m + • can be lifted to that of V m +1+ • as is notedin Remark (1.4). We have the last assertion of Theorem 1 since the δ k ( k ≥
0) underlieisomorphisms of mixed Hodge structures. We thus get the first proof of Theorem 1 in thispaper.
2. Geometric proof of Theorem 1
In this section we give two proofs of Proposition 2, and then a geometric proof of Theorem 1after showing Lemma (2.3).
Consider the following morphisms of mixed Hodgemodules on S :(2 . . M ! := H f ∗ ( j ! Q h,X [ n ]) u ′ → H f ∗ IC X Q h v ′ → M ∗ := H f ∗ ( j ∗ Q h,X [ n ]) . These are induced by the canonical morphisms of mixed Hodge modules on X whose under-lying morphisms are as in (1.1.2):(2 . . j ! Q h,X [ n ] u → IC X Q h v → j ∗ Q h,X [ n ] . By (1.1.4) the kernel and cokernel of u ′ and v ′ are constant mixed Hodge modules on S .By the formalism of mixed Hodge modules (see e.g. [Sai2], 2.26) we have(2 . .
3) Gr Wn + k ( j ! Q h,X [ n ]) = Gr Wn − k ( j ∗ Q h,X [ n ]) = 0 if k > , and moreover(2 . .
4) Gr Wn ( j ! Q h,X [ n ]) = Gr Wn ( j ∗ Q h,X [ n ]) = IC X Q h . (Indeed, for the last assertion, we use Hom( M ′ , j ∗ Q h,X ) = Hom( j ∗ M ′ , Q h,X ) = 0 for anymixed Hodge module M ′ supported on X \ X , and similarly for the dual assertion.) Note thatthe weight filtration W on M ! and M ∗ is induced by the weight filtration W on j ! Q h,X [ n ]and j ∗ Q h,X [ n ] respectively via the weight spectral sequence, see [Sai2], Prop. 2.15. Moreover,this W induces the weight filtration W on H s , H cs if s is in a sufficiently small non-emptyZariski-open subset of S (since W is independent of the choice of a compactification). EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 7
By (1.1.4) we have constancy of the kernel and the cokernel of the canonical morphism(2 . .
5) Gr Wn M ! → Gr Wn M ∗ . Assume the cokernel is nonzero. We have Gr Wk M ∗ = 0 for k < n by (2.1.3). So there isa nontrivial constant Hodge submodule in M ∗ by semisimplicity of pure Hodge modulesapplied to Gr Wn M ∗ . (The latter property follows from polarizability of pure Hodge modulesby using [Sai2], Th. 3.21.) However, this contradicts the property that H − ( S, M ∗ ) = 0which follows from the condition that X = C n by using the Leray spectral sequence as inRemark (1.2). So we get the surjectivity. For the injectivity we apply the dual argument.Restricting over a sufficiently general s ∈ S , we then get the desired isomorphism. Set Y = X s . More generally, let Y be a smoothvariety which is the complement of an ample effective divisor E on a projective variety Y .Under this assumption, we show the bijectivity of the canonical morphism(2 . . Gr Wm H mc ( Y ) → Gr Wm H m ( Y ) . We have a smooth projective compactification e Y of Y such that D := e Y \ Y is a divisor withsimple normal crossings. This is obtained by using Hironaka’s resolution σ : ( e Y , D ) → ( Y , E )which is a projective morphism. Let D σ be a relatively ample divisor for σ . We may replace D σ with D σ − σ ∗ σ ∗ D σ so that its support is contained in D . Then kσ ∗ E + D σ is an ampledivisor on e Y for k ≫
0. Since its support is contained in D , it is a linear combination of theirreducible components D i of D .By Deligne’s construction of mixed Hodge structure on H i ( Y ) (see [De1]) together withduality, we have(2 . . Gr Wm H mc ( Y ) = Ker (cid:0) H m ( e Y ) → L i H m ( D i ) (cid:1) ,Gr Wm H m ( Y ) = Coker (cid:0)L i H m − ( D i )( − → H m ( e Y ) (cid:1) . So the assertion is equivalent to the non-degeneracy of the restriction of the natural pairingon the middle cohomology H m ( e Y ) to the kernel of the morphism H m ( e Y ) → L i H m ( D i ). ByHodge theory, it is enough to show that this kernel is contained in the primitive part withrespect to the above ample divisor. But it is clear since the action of the cohomology classof each D i is given by composing the restriction morphism H • ( e Y ) → H • ( D i ) with its dual.So the assertion follows. This finishes another proof of Proposition 2.For the geometric proof of Theorem 1 in this section, we also need the following. Lemma 2.3.
Let H be a mixed Hodge structure, and L an increasing filtration on H . Let N be a nilpotent endomorphism of type ( − , − of H preserving the filtration L . Assume therelative monodromy filtration W for ( L, N ) exists, and W coincides with the weight filtrationof the mixed Hodge structure H . Let m be an integer such that H = L m H . Assume theaction of N on L m − H vanishes, and (2 . .
1) dim Gr Wm − k (Ker N ) = dim Gr Wm + k (Coker N ) ( k ≥ , Gr Wm − k (Ker N ) = Gr Wm + k (Coker N ) = 0 ( k ≤ , where Coker N is a quotient of H ( − , and Ker N ⊂ H . Then W coincides with the mon-odromy filtration with center m − . ALEXANDRU DIMCA AND MORIHIKO SAITO
Proof.
Set H ′ := L m − H , H ′′ := Gr Lm H . The hypothesis on the action of N on H ′ impliesthat(2 . .
2) Gr Wk Gr Li H ′ = 0 ( k = i ) , i.e. W = L on H ′ . Set H k := Gr Wk H , and similarly for H ′ k , H ′′ k . Set K k := Ker (cid:0) N : H k → H k − ( − (cid:1) , C k := Coker (cid:0) N : H k → H k − ( − (cid:1) , and similarly for K ′′ k , C ′′ k . Note that Gr Wk commutes with Ker and Coker by the strictcompatibility of the weight filtration W .Applying the snake lemma to the action of N on 0 → H ′ → H → H ′′ →
0, we get thefollowing long exact sequence for any k ∈ Z (2 . .
3) 0 → H ′ k → K k → K ′′ k ∂ → H ′ k − ( − → C k → C ′′ k → . Here H ′ k = K k = 0 for k ≥ m , and C k = 0 for k ≤ m by hypothesis.We show the following isomorphisms by decreasing induction on k ≥ . . H ′ m − k ∼ −→ K m − k , ∂ : K ′′ m − k ∼ −→ H ′ m − k − ( − . Here it is enough to show that dim K ′′ m − k = dim H ′ m − k − , using the long exact sequence(2.3.3) since the surjectivity of ∂ follows from the vanishing of C k − for k ≤ m .For k ≫
0, the assertion trivially holds since all the terms are zero. Assume the isomor-phisms hold with k replaced by k + 2. We have the following equalities for k ≥ K ′′ m − k = dim C ′′ m + k +2 = dim C m + k +2 = dim K m − k − = dim H ′ m − k − . Indeed, the first equality follows from the property of the monodromy filtration on H ′′ , thesecond from the long exact sequence (2.3.3) together with the hypothesis that H ′ k − = 0for k ≥ m + 2, the third from the hypothesis (2.3.1) of the lemma, and the last from theinductive hypothesis. So the two isomorphisms in (2.3.4) hold for k ≥ H and the dual filtration of L on it.Then (2.3.4) implies that the primitive decomposition of L k H ′′ k with center m can be liftedto the primitive decomposition of L k H k with center m − H → H ′′ ,since K ′′ k is the coprimitive part of H ′′ k ( k ≤ m ). This finishes the proof of Lemma (2.3). We show the assertion for H c ∞ since that for H ∞ follows fromthis using duality. Consider first the following canonical morphisms R Γ c ( S, f ! Q h,X ) α −→ R Γ( S, f ! Q h,X ) β −→ R Γ( S, f ∗ Q h,X ) . Set γ = β ◦ α : R Γ c ( S, f ! Q h,X ) → R Γ( S, f ∗ Q h,X ) . By the octahedral axiom of the derived category, we get a distinguished triangle(2 . . C (cid:0) α ) → C ( γ ) → C ( β ) +1 → . By (1.1) the following mapping cone is a direct sum of constant sheaves on S : C ( f ! Q h,X → f ∗ Q h,X (cid:1) = C (cid:0) R f ∗ j ! Q h,X → R f ∗ j ∗ Q h,X (cid:1) , EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 9 and this holds in D b MHM( S ). Moreover, the stalk at s ∈ S ′ of the mapping cone is givenby the cohomology of the mapping cone C (cid:0) R Γ c ( X s , Q ) → R Γ( X s , Q ) (cid:1) ( s ∈ S ′ ) , using the generic base change by the inclusion { s } ֒ → S .We then get the following isomorphisms in the derived category of graded-polarizablemixed Hodge structures D b MHS:(2 . . C ( α ) = C (cid:0) N : ψ /t, f ! Q h,X → ψ /t, f ! Q h,X ( − (cid:1) [ − ,C ( β ) = C (cid:0) R Γ c ( X s , Q ) → R Γ( X s , Q ) (cid:1) ( s ∈ S ′ ) ,C ( γ ) = Q ⊕ Q ( − n )[3 − n ] . Indeed, the first isomorphism follows from C ( j ′ ! M → j ′∗ M ) = C (cid:0) N : ψ /t, M → ψ /t, M ( − (cid:1) [ − , for any mixed Hodge module M on S where j ′ : S ֒ → S := P is the inclusion, see [Sai2],2.24. (In this paper the nearby and vanishing cycle functors ψ , ϕ for mixed Hodge modulesare compatible with those for the underlying Q -complexes in [De2], [Di2] without any shift ofcomplexes, and do not preserve mixed Hodge modules.) The second isomorphism of (2.4.2)follows from the above argument on the mapping cone, and the last isomorphism of (2.4.2)from R Γ c ( X, Q ) = R Γ c ( S, f ! Q h,X ) , R Γ( X, Q ) = R Γ( S, f ∗ Q h,X ) . Set m = n −
1. With the notation in the main theorem, we have the decompositions R c Γ( X s , Q ) ∼ = H cs [ − m ] ⊕ Q ( − m )[ − m ] , R Γ( X s , Q ) ∼ = H s [ − m ] ⊕ Q , using the vanishing of Ext i ( i >
1) in MHS together with the filtration τ ≤ k as above. Wealso have ψ /t, f ! Q h,X ∼ = H c ∞ , [ − m ] ⊕ Q ( − m )[ − m ] . Let ι s : H cs → H s denote the canonical morphism. The distinguished triangle (2.4.1) is thenequivalent to the isomorphism in D b MHS:(2 . . C (cid:0) N : H c ∞ , → H c ∞ , ( − (cid:1) ∼ = C (cid:0) ι s : H cs → H s (cid:1) . Note that Ker ι s and Coker ι s for s ∈ S ′ are extended to constant variations of mixed Hodgestructures over S by (1.1).By duality we have(2 . . D (cid:0) Gr Wm − k H cs (cid:1) = (cid:0) Gr Wm + k H s (cid:1) ( m ) for k ≥ . Since X s is smooth affine, we haveGr Wm − k H cs = Gr Wm + k H s = 0 for k < . This implies that Gr Wm + k ι s = 0 ( k = 0), and Gr Wm ι s is an isomorphism by Proposition 2.Combining these with the isomorphism (2.4.3) in D b MHS, we get(2 . .
5) Gr Wm + k (Ker N ) ∼ = ( Gr Wm + k H cs if k < , k ≥ , Gr Wm + k (Coker N ) ∼ = ( Gr Wm + k H s if k > , k ≤ , where Coker N is a quotient of H c ∞ , ( − D (cid:0) Gr Wm − k (Ker N ) (cid:1) = (cid:0) Gr Wm + k (Coker N ) (cid:1) ( m ) for k > . Then, applying Lemma (2.3) to H = H c ∞ , where L k H is identified with W k H cs for s ∈ S ′ ,we get the second proof of Theorem 1 in this paper.
3. Universal extensions by constant sheaves
In this section we explain the notion of a universal extension by a constant sheaf, and thenprove Theorem 3 which implies Theorem 1.
Let M be any pureHodge module of weight n on S = C having no constant direct factors, i.e. H − ( S, M ) = 0.Note that H j ( S, M ) = 0 for j > S is affine.Consider the functor E M ( H ) := Ext S ) ( H S [1] , M ) for H ∈ MHS , where H S := a ∗ S H with a S : S → pt the projection. Set H M := H ( S, M ) , ( H M ) S := a ∗ S H M . Since a ∗ S is the left adjoint functor of ( a S ) ∗ , we get the first isomorphism of the functorialcanonical isomorphisms(3 . . E M ( H ) ∼ −→ Ext ( H [1] , ( a S ) ∗ M ) = Hom MHS ( H, H M ) , where the second isomorphism follows from the vanishing of H j ( S, M ) for j = 0. Note thatthe first isomorphism is given by taking the direct image of u : H S [1] → M by a S , and thencomposing it with the adjunction morphism H [1] → ( a S ) ∗ a ∗ S H [1].By (3.1.1), E M is represented by H M . This means that there is a universal extension f M of M by a constant mixed Hodge module on S :(3 . .
2) 0 → M → f M → ( H M ) S [1] → , whose extension class corresponds to the identity on H M by the isomorphism (3.1.1), andsuch that any extension class ξ ∈ Ext S ) ( H S [1] , M ) is obtained by taking the pull-backof the short exact sequence (3.1.2) by the morphism( v ξ ) S : H S [1] → ( H M ) S [1] , which is induced by a morphism v ξ : H → H M in MHS, where v ξ is uniquely determined bythe extension class ξ . EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 11
Lemma 3.2.
Let H := ψ /t, M , e H := ψ /t, f M so that we have the exact sequence in MHS :(3 . .
1) 0 → H → e H → H ( S, M ) → . By the action of N := (2 πi ) − log T u together with the diagram of the snake lemma, we havethe morphism (3 . . ∂ ′′ : H ( S, M ) → Coker( N | H ) , where Coker( N | H ) is a quotient of H ( − . Assume the following condition holds :( C ) ∂ ′′ is surjective and Ker ∂ ′′ = W n H ( S, M ) .Then the weight filtration W on H coincides with the monodromy filtration shifted by n .Proof. This follows from the primitive decomposition as in the proof of Lemma (1.3).
By Lemma (3.2) above we have to prove condition (C). It isenough to show this condition for the underlying perverse sheaves. Let F be the underlying Q -perverse sheaf of M . Set E F ( V ) := Ext S ) ( V S [1] , F ) for V ∈ M f ( Q ) , where M f ( Q ) denotes the category of finite dimensional Q -vector spaces. By a similarargument, this functor is also represented by H ( S, F ) = H ( S, M ) Q .Let ∆ be a sufficiently small open disk in P with center ∞ such that F ∆ ∗ is a local systemup to a shift where ∆ ∗ := ∆ \ {∞} . Define for V ∈ M f ( Q ) E F ∆ ∗ ( V ) := Ext ∗ ) ( V ∆ ∗ [1] , F ∆ ∗ ) = Hom Q ( V, H (∆ ∗ , F ∆ ∗ )) . Set H , Q := ψ /t, F [ − E F ∆ ∗ is represented by H (∆ ∗ , F ∆ ∗ ) = Coker( N : H , Q → H , Q ( − . We have the canonical functor morphism E F → E F ∆ ∗ , which corresponds to the canonical morphism(3 . . H ( S, F ) = H ( S, R j ∗ F ) → H (∆ ∗ , F ∆ ∗ ) = H (∆ , ( R j ∗ F ) | ∆ ) , where j : S ֒ → S = P . We have to calculate the morphism (3.3.1).Let W be the weight filtration on R j ∗ F . By the proof of [Sai2], Prop. 2.11 (see also [StZ]),we have(3 . .
2) Gr Wk ( R j ∗ F ) = k < n,j ! ∗ F if k = n,i ∗ Gr Wk Coker( N | H , Q ) if k > n, where N : H , Q → H , Q ( −
1) is as above, and i : {∞} ֒ → S is the inclusion. Since H ± ( S, j ! ∗ F ) = 0 by hypothesis and duality, we getGr Wk H ( S, F ) = H ( S, Gr Wk ( R j ∗ F )) . Let j ′ : ∆ ∗ ֒ → ∆ denote the inclusion so that j ! ∗ F | ∆ = j ′ ! ∗ ( F ∆ ∗ ) . By the local classification of perverse sheaves on ∆ (see e.g. [BdM], [BrMa]), we have(3 . .
3) Ext ( V ∆ [1] , j ′ ! ∗ ( F ∆ ∗ )) = 0 , and furthermore(3 . . E F ∆ ∗ ( V ) = Ext ( V ∆ [1] , R j ′∗ ( F ∆ ∗ ))= Ext ( V ∆ [1] , i ′∗ Coker( N | H , Q ))= Hom Q ( V, Coker( N | H , Q )) , where i ′ : {∞} ֒ → ∆. Let e F and ] F | ∆ ∗ respectively be the universal extensions of F and F ∆ ∗ by constant perverse sheaves so that we have the short exact sequences in Perv( S ) andPerv(∆ ∗ ): 0 → F → e F → H ( S, F ) S [1] → , → F ∆ ∗ → ] F | ∆ ∗ → Coker( N | H , Q ) ∆ ∗ [1] → . By (3.3.3–4) we have the following commutative diagram of exact sequences in Perv(∆ ∗ ):(3 . .
5) 0 0 ↓ ↓ → H ( S, j ! ∗ F ) ∆ ∗ [1] → H ( S, j ! ∗ F ) ∆ ∗ [1] → ↓ ↓ ↓ → F ∆ ∗ → e F (cid:12)(cid:12) ∆ ∗ → H ( S, F ) ∆ ∗ [1] → || ↓ ↓ → F ∆ ∗ → ] F | ∆ ∗ → Coker( N | H , Q ) ∆ ∗ [1] → ↓ ↓ ↓ Wn H ( S, F ) = H ( S, j ! ∗ F ). Indeed, the assertion is equivalent to that the quotientof the middle row by the top row is isomorphic to the bottom row. By (3.3.3–4) this followsfrom the fact that the restriction to ∆ induces the isomorphism of extension classes:Ext S ) ( V S [1] , i ∗ Coker( N | H , Q )) ∼ −→ Ext ( V ∆ [1] , i ′∗ Coker( N | H , Q )) . Note that the morphism ∂ ′′ in Lemma (3.2) is functorially defined for any short exactsequences on ∆ ∗ whose last term is constant, and it is bijective in the case of the short exactsequence associated to the local universal extension ] F | ∆ ∗ . So the assertion follows from theabove commutative diagram of short exact sequences. With the notation of (3.1), let δ : S ֒ → S × S be the diagonal, and q i : S × S → S the i -th projection ( i = 1 , v : H [1] → ( a S ) ∗ M by a S and then composing it with the functorial morphism: a ∗ S ( a S ) ∗ M = ( q ) ∗ q ∗ M → ( q ) ∗ δ ∗ δ ∗ q ∗ M = M . Let j S denote the inclusion of the complement of δ ( S ) in S × S . Then we have the distin-guished triangle in D b MHM( S )( q ) ∗ ( j S ) ! j ∗ S q ∗ M → ( q ) ∗ q ∗ M → M +1 → , EIGHT FILTRATION OF LIMIT MIXED HODGE STRUCTURE 13 and it gives the short exact sequence (3.1.2) in MHM( S ) together with the isomorphism(3 . . f M = ( q ) ∗ ( j S ) ! j ∗ S q ∗ M [1] . This is essentially the same as the definition of f M by C. Sabbah in Appendix of [MT].Consider the long exact sequence associated to (3.1.2):0 → H − ( S, f M ) → H M ∂ ′ → H ( S, M ) → H ( S, f M ) → , where H − ( S, M ) = 0 by hypothesis and H j ( S, M ) = H j ( S, f M ) = 0 for j > S isaffine. Here ∂ ′ is the identity (up to a sign) by the definition the first isomorphism in (3.1.1).So we get(3 . . H j ( S, f M ) = 0 for any j ∈ Z . Conversely, if there is a short exact sequence(3 . .
3) 0 → M → M ′ → H ′ S [1] → , with H ′ ∈ MHS and M ′ ∈ MHM( S ) satisfying the vanishing condition as in (3.4.2), then M ′ is identified with the universal extension f M of M by a constant mixed Hodge moduleon S . Indeed, this follows by applying the functor on the right-hand side of (3.4.1) to theshort exact sequence (3.4.3), since M ′ = f M ′ by the vanishing condition on H j ( S, M ′ ) andthe right-hand side of (3.4.1) vanishes for a constant Hodge module on S .Since the vanishing condition as in (3.4.2) is satisfied for M ′ = H f ∗ ( Q h,X [ n ]) (using theLeray spectral sequence as in Remark (1.2)), we get(3 . . f M = H f ∗ ( Q h,X [ n ]) with M = Gr Wn H f ∗ ( Q h,X [ n ]) . So Theorem 3 implies the third proof of Theorem 1 in this paper.
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