The Perverse Filtration and the Lefschetz Hyperplane Theorem

Home , Leray spectral sequence, Perverse sheaf

The perverse ﬁltration and the Lefschetz Hyperplane Theorem

Mark Andrea A. de Cataldo∗ and Luca Migliorini†

April 2007

Abstract We describe the perverse ﬁltration in cohomology using the Lefschetz Hyperplane Theorem.

Contents

1 Introduction 2

2 Notation and preliminaries 4

3 The spectral sequences associated with a flag and with the perverse t−structure 7 3.1 Spectral sequences: the F and the L filtration, re-numeration, translation, shift ...... 7 3.2 The diagonal filtration ...... 8 3.3 The flag spectral sequences ...... 8 3.4 The perverse spectral sequence ...... 10 3.5 The Jouanolou Trick ...... 11

4 Resolutions in derived category of perverse sheaves 12 4.1 Beilinson Equivalence Theorem ...... 12 4.2 Cohomology as a left derived functor ...... 14 4.3 R0−acyclic ﬁltered resolutions ...... 14 4.4 Double complexes: the ﬁltrations F1 and F2 := Dec(F1) ...... 15 4.5 Resolutions of perverse sheaves via hyperplane sections ...... 16 b 4.6 Resolutions in C (PY ) using hyperplane sections ...... 18

∗Partially supported by N.S.F. †Partially supported by GNSAGA

1 5 The geometry of the perverse ﬁltration 19 5.1 The statement of the main result ...... 19 5.2 Two lemmas on biﬁltered complexes ...... 20 5.3 Construction of an isomorphism of spectral sequences ...... 22 5.4 Proof of Theorem 5.1 ...... 24

6 Some Hodge-theoretic applications 24

1 Introduction secintro In this paper we use hyperplane sections to give a geometric description of the middle perverse filtration on the cohomology of a constructible complex on a quasi projective variety. This description is somewhat surprising, especially if one views the constructions leading to perverse sheaves as transcendental and hyperplane sections as more algebro- geometric. Let Y be an affine variety of dimension n and K be a constructible complex on Y . By the theorem on the cohomological dimension of affine varieties, for every perverse sheaf P on Y, the cohomology groups Hj(Y,P ) = 0, for j∈ / [−n, 0]. It follows that the perverse filtration L pτ in cohomology, i.e. the one stemming from the perverse spectral sequence, −n 1 satisfies L pτ H = H and L pτ = 0, i.e. it is of type [−n, 0]. A n−flag Y∗ on Y is a sequence of closed subvarieties Y−n ⊆ ... ⊆ Y0 = Y. The classical p+q p+q flag spectral sequence H (Yp,Yp−1,K) =⇒ H (Y,K) has associated filtration FY∗

given by the kernels of the restriction maps H(Y,K) → H(Yp,K|Yp ). The filtration FY∗ is of type [−n, 0]. The main result of this paper is that when Y is affine and Y−i is the intersection of i general hyperplane sections of Y, the two filtrations become identical:

p L τ = FY∗ ,

i.e. the perverse filtration is obtained using the restriction maps. More is true: the flag and perverse spectral sequences coincide. By dualizing, we obtain the same result for cohomology with compact supports. The flag Y∗ must be cellular for the perverse cohomology of the given complex K in the sense that the relative cohomology of the scheleta with coefficients in the perverse cohomology is non zero at most in one prescribed degree. This is in analogy with CW- complexes. The important difference is that the scheleta are algebraic subvarieties. The case of Y quasi projective is handled by reducing to the affine case by a trick of Jouanolou’s. A lengthier, yet more natural, proof can be given y using a Cech-type argument involving finite affine coverings. Our methods seem to break down in the non quasi projective case and also for other perversities. The constructions and results are amenable to mixed Hodge theory.

2 We offer the following two applications: Let f : X → Y be any map, Y quasi projective and K be any complex n X. Then the perverse Leray spectral sequences can be identified with suitable flag spectral sequences on X. In the special case when K = QX , the perverse spectral sequences for H(X, Q) and Hc(X, Q), are spectral sequences of mixed Hodge structures. We endow the intersection cohomology groups of a quasi projective variety with a natural mixed Hodge structure which is compatible with: resolution of singularities, the natural map from ordinary cohomology, the Poincarépairing in intersection cohomology, the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. This second set of applications requires the methods of [5] and of [6]. In the interest of the self- containedness of this paper, the details will appear elsewhere. Some of these applications are already afforded by M. Saito’s theory of mixed Hodge modules. We believe that the simple-minded approach to these applications can lead to useful insight into geometric questions. Arapura’s paper [1] deals with the standard versus the perverse, filtration. For the flag to be cellular for the ordinary cohomology sheaves, the hyperplane sections must contain the bad loci of these sheaves. The standard, i.e Grothendieck’s, spectral sequence is related to any flag spectral sequence as follows. Since the trivial filtration is coarser than the flag filtration, and the the standard filtration is the shift of the trivial filtration, if follows that the standard filtration is coarser than the shifted flag filtration. This gives a map from the standard spectral sequence to the shifted flag spectral sequence. However, there is no relation between the latter and the flag spectral sequence; e.g. as in the case of the trivial and standard filtrations. This is because shifting and resolving do not commute. The use of Godement resolutions allows to use a isomorphic model of K for which the relation at the level of complexes between a filtration and its shift is reproduced exactly in the cohomology spectral sequences. This implies, using the isomorphic model, the existence of a map from the standard spectral sequence to the flag spectral sequence. A flag which is cellular for the cohomology sheaves yields an isomorphism. The situation is quite different for the perverse spectral sequence, i.e. the object of this paper: the perverse truncation functors do not appear to arise in connection with a shift of filtration operation in the derived category of sheaves. Our solution to this problem is to work with perverse sheaves instead of of sheaves. Beilinson has proved that the bounded derived category of perverse sheaves is equivalent to the bounded-derived-type category of complexes with constructible cohomology. This equivalence transforms the standard truncation in the former category into the perverse truncation in the latter and, by working with perverse sheaves we can view perverse truncation as an ordinary truncation. However, the category of perverse sheaves does not have enough injectives. Let Y be affine and K1 be a bounded complex of perverse sheaves corresponding to the complex of sheaves K via the equivalence. We show that the choice of a general flag Y∗, adapted to K1 yields a complex SY∗ (K1) of perverse sheaves endowed with two filtrations F1 and F2. These filtrations have several properties. There is an

3 acyclic filtered resolution (SY∗ (K1),F2) of K1 endowed with the standard filtration. Via Beilinson equivalence, this translates into the presentation of an explicit filtered complex • 0 (V ,F2) of vector spaces computing the perverse spectral sequence for K. The filtration • 0 F1 stems from the flag Y∗ and yields (V ,F1), which is, up to natural isomorphism, the flag spectral sequence for K. The remarkable fact, is that the shift of F1 is F2. This fact has no counterpart for ordinary sheaves, where the shift produces a “coarser-than”-type 0 0 relation. By acyclicity, we also have that the shift of F1 is F2, and we deduce the existence of the wanted isomorphism between the flag and perverse spectral sequences.

Acyclicity means that the flag is cellular for all the perverse entries of SY∗ (K1) and related subquotients. This is achieved by a systematic use of the Lefschetz Hyperplane Theorem of Goresky and MacPherson. In our paper [5], we proved that the perverse filtration on IH(X) associated with a projective map of projective varieties f : X → Y agreed with the canonical filtration ∗ associated with the nilpotent operator c1(f (A)), where A is any fixed ample line bundle on Y, acting on IH(X) by cup product. This fact is equivalent to a series of Hard Lefschetz-type results on the perverse graded pieces of IH(X) and it implies that the perverse filtration is by pure Hodge structures. This is the cornerstone to our approach to the proof of the Decomposition Theorem in [5]. This approach to the perverse filtration does not generalize to non compact varieties, for line bundles can be simultaneously ample and have trivial Chern class. This paper provides a universal description of the perverse filtration in cohomology, using not the Hard Lefschetz, but the Weak Lefschetz Theorem.

The paper is organized as follows. Section 2 sets up the notation and recalls the facts needed from the theory of perverse sheaves. Section 3 defines the perverse and the flag spectral sequences. Section 4 constructs suitable resolutions, Proposition 4.9, in the derived category of perverse sheaves. Section 5 contains the statement and proof of the main result, Theorem 5.1, identifying the flag and perverse spectral sequences. Section 6 contains the aforementioned applications to mixed Hodge theory.

2 Notation and preliminaries secnotprel In this paper a variety is a separated scheme of finite type over C. A map of varieties is a C−morphism of varieties. The stratifications employed are complex algebraic. Any map can be stratified. Let Y be a variety. The category DY is the full subcategory of the bounded derived category of sheaves of Q−vector spaces, given by complexes with constructible cohomology. The finite dimensional hypercohomology Q−vector spaces are denoted H(Y,K) and p ≤0 Hc(Y,K). The category DY is endowed with the middle perversity t−structure DY , p ≥0 p p p i p ≤0 p ≥0 DY , τ≤i, τ≥i, H ,... The heart PY = DY ∩ DY is the abelian category of middle perversity perverse sheaves on Y. We omit mentioning “middle.”

4 Σ Σ Let Σ be a stratification of Y. We have the full subcategories DY , PY of Σ−constructible Σ complexes. The category PY is abelian. ∨ The Verdier dual of a complex K ∈ DY is denoted K . The duality functor preserves stratifications. There is the canonical Poincaré-Verdier Duality isomorphism: Hi(Y,K∨) ' −i ∨ Hc (Y,K) . ∗ ! ∗ ! Given a map f : X → Y, the usual functors (f , Rf∗, Rf!, f ) are denoted (f , f∗, f!, f ). 0 Σ Σ0 If f :(X, Σ) → (Y, Σ ) is a stratified map, the f∗, f! : DX → DY , etc. All filtrations F are decreasing, i.e. F i ⊇ F i+1 and finite, i.e. F l = Id and F m = 0, for some l ≤ m. In this case, we say that F is of type [l, m]. In the remainder of this section, we collect some useful facts about perverse sheaves. Let j U −→ Y ←−i D be such that j is an open immersion and i is the complementary closed immersion. Given 0 (U, ΣU ), there is a stratification (Y, Σ ) such that D and U are union of strata, the maps 0 0 0 0 i and j are stratified for Σ and the induced stratifications ΣU and ΣD, and ΣU refines Σ. ! ∗ + There is the distinguished triangle functor j!j → Id → i∗i → . The intermediate extension functor j!∗; PU → PY is defined via the canonical factorization diagram p 0 ! epi mono p 0 ∗ H (j!j (−)) −→ j!∗(−) −→ H (j∗j (−)).

This functor is seldom exact. The intermediate extension j!∗P of P ∈ PU to PY is characterized by not admitting any nontrivial subquotient supported on D. Let Y ⊇ S ∈ Σ be irreducible and L be a local system on S. If j : S → S ← S \ S : i,

define the intersection cohomology complex IS(L) := j!∗L[dim S] ∈ PS. It can be viewed in PY via the fully faithful functor i∗. If L is an irreducible local system on S, the IS(L) is a simple object in PY . Viceversa, every simple object of PY has this form for some (S, L), where S ⊆ Y a smooth irreducible subvariety and L an irreducible local system on S. The category PY is Noetherian and Artinian. Every perverse sheaf P ∈ PY admits 0 l i a filtration F P ⊆ ...F P ⊆ 0 with the property that every graded piece grF P is simple. The non-trivial graded pieces are called the constituents of P. The filtration is not canonical. However, the set of constituents is and its cardinality l(P ) is called the length of P. Σ A simple induction on the length shows that if P ∈ PY , then so are the constituents and any subquotient of P. Assume that the open immersion j : U → Y is affine, e.g. D is a Cartier divisor. Then the functors j!, j∗ are t−exact, i.e., in this bounded context, preserve perverse sheaves. It follows automatically that j!, j∗ : PU → PY are exact.

Σ prtex Lemma 2.1 Let P ∈ PY . Assume that j is aﬃne and that D does not contain any of the strata of Σ. Then 1) There is the natural exact sequence in PY :

∗ ! 0 −→ i∗i P [−1] −→ j!j P −→ P −→ 0.

5 2) Let Σ0 be any reﬁnement of Σ such that D is a union of strata of Σ0. Then the functors

! ∗ Σ Σ0 j!j , i∗i [−1] : PY −→ PY ⊆ PY are exact.

∗ ! + Proof. (1). Consider the distinguished triangle i∗i P [−1] → j!j P → P → . Since j! and ! ∗ ! ! j = j are t−exact, then so is j!j and j!j P ∈ PY . The associated long exact sequence of p r ∗ perverse cohomology shows that H (i∗i P [−1]) = 0, for every r 6= −1, 0 and yields the exact sequence in PY :

p −1 ∗ a ! b 0 −→ H (i∗i P [−1]) −→ j!j P −→ P −→ 0. We need to show that b is epi. Assume ﬁrst that P is simple. Then P has no non-trivial subquotients supported D. It follows that b is epi, in this case. The fact that b is epi in the general case follows by induction on the length l(P ) of P and the commutative diagram of short exact sequences

1 0 0 / F P / P / grF P / 0

epi b epi ! 1 ! ! 0 0 / j!j F P / j!j P / j!j grF P / 0. This establishes (1). 0 0 0 Σ Since i and j are stratiﬁed with respect to Σ , Σ|D and Σ|U , we have that for every P ∈ PY , ∗ ! Σ Σ0 i∗i , j!j : PY → PY . The exactness statement follows by applying each functor to any short exact sequence and noting that, by part (1), there is no non zero perverse cohomology around.

The following results are due to Goresky and MacPherson, are valid with Z−coefficients and have been proved using Stratified Morse Theory. cdav Theorem 2.2 (Cohomological dimension of affine varieties.) Let Y be affine, P ∈ PY . Then Hr(Y,P ) = 0, ∀r > 0.

swl Theorem 2.3 (Strong Weak Lefschetz) Let Y be aﬃne and P ∈ PY . Let Y ⊆ P be any embedding in projective space. Let H ⊆ Y be a general hyperplane section, with corresponding open immersion j : Y \ H → Y. Then

r ! H (Y, j!j P ) = 0, ∀r 6= 0. serve Remark 2.4 The hyperplane H can be chosen as follows. Let Y ⊆ Y ⊆ P be a projective completion. Let Σ be a stratiﬁcation of Y such that H∞ := Y \ Y is a union of strata and such that P is ΣU −constructible. Let H ⊆ Y be a hyperplane section of meeting the strata on H∞ transversally. Then H can be chosen to be the trace on Y of H.

6 3 The spectral sequences associated with a flag and with the perverse t−structure spefl 3.1 Spectral sequences: the F and the L filtration, re-numeration, translation, shift ssecss b b Let (K,F ) be a filtered complex in DY , (or D (PY ), or D (VQ)), and T : DY → VQ be a cohomological functor. Verdier’s formalism of spectral objects works in the absence of enough T −acyclic objects and yields a spectral sequence

pq p+q E1 (K,F ) =⇒ T (K),

with abutment

0p j j p j 0p p+q 0p+1 p+q pq F T = Im (T (F K) → T (K)),F T /F T = E∞.

It is customary to denote F 0 by F. The re-numeration Er+1 of Er, is the same spectral sequence, with a diﬀerent indexing scheme:

st −t,s+2t st s+t Er+1(K,F ) := Er (K,F ), r ≥ 1, E (Y,K) =⇒ T (K). (1) ren

One still has the ﬁltration F 0, but also has the L−ﬁltration

s j 0s+j j s s+t s+1 s+t st L T := F T ,L T /L T = E∞. (2) lf

grren Example 3.1 If one re-numbers the usual Grothendieck spectral sequence for hyper- pq p+q −p p+q 0p j cohomology E1 = H (Y, H (K)[p]) =⇒ H (Y,K), which has F H (Y,K) = st s t s j Im(H(Y, τ≤−pK) → H(Y,K)), then one obtains E2 = H (Y, H (K)) with L H (Y,K) = j Im(H (Y, τ≤−s+j(K)).

The l−translate G(l) of a filtration G is defined by G(l)s := Gl+s. The l−translate Er(l) of a spectral sequence Er with abutment a filtration G is defined as follows: pq p+l,q−l p+q Er(l) := Er =⇒ T . and the associated abutted filtrations satisfy,

G(E1(l)) = G(E1)(l). (3) trfi

This formula applies to the F 0 filtration for (K,F ) and clearly holds as well for the associated L filtration. Let (K,F ) be a filtered complex of abelian groups. The shifted filtration Dec(F ) of K is defined by: Dec(F )pKm := {x ∈ F p+mKm | dx ∈ F p+m+1Km+1} (4) decf

7 This deﬁnition can be adapted to ﬁltered complexes in any abelian category. Two important properties are the following equality taking place in the cohomology H(K) of the complex p p+m Dec(F )0 Hm(K) = F 0 Hm(K), (5) decfs and that there is a natural isomorphism of spectral sequences

pq 2p+q,−p Er (K, Dec(F )) ' Er+1 (K,F ). (6) decss

Note that (5) does not hold after the application of a cohomological functor, e.g. hypercohomology. The last property should not be confused with the act (1) of re-numbering the spectral sequence on the lhs. If we re-number the lhs, then we obtain

st st s 0 0s Er+1(K, Dec(F )) ' Er+1(K,F ), r ≥ 1 L (Dec(F ) ) = F . (7) declf

These identiﬁcations are crucial in this paper: the lhs will play the role of the re-numbered perverse spectral sequence, while the rhs will play the role of the spectral sequence associated with a sequence of closed subspaces and the two are “one diﬀerential apart.”

3.2 The diagonal filtration diafi If a complex K is endowed with two filtrations F and G, we can construct the diagonal filtration δ(F,G), see [3] 3.1.2.8, as follows:

δ(F,G)k = X F a ∩ Gb a+b=k The graded pieces of δ(F,G) are:

k M a b grδ(F,G) = grF grG. a+b=k

This ﬁltration will play an important role in the proof of our main result.

3.3 The flag spectral sequences ssflag In this section, we define what we mean by flag spectral sequences. A n−flag Y∗ on the variety Y is defined to be a sequence

∅ = Y−n−1 ⊆ Y−n ⊆ ... ⊆ Y−1 ⊆ Y0 = Y (8) nflag

of closed subvarieties of Y. The indexing scheme is chosen in order to obtain ﬁltrations FY∗ of type [−n, 0] in cohomology and of type [0, n] in cohomology with compact supports. Let I−s,−t : Y−s \ Y−t → Y, 0 ≤ s ≤ t ≤ n + 1

8 be the locally closed embeddings. The functors

∗ (−)−s,−t := (I−s,−t)!(I−s,−t) : DY → DY

are just restrictions, followed by extension by zero. Let ip := Ip,−n−1 : Yp → Y be the closed embedding. Let K ∈ DY . There is the chain of inclusions

0 ⊆ K0,−1 ⊆ ... ⊆ K0,−n ⊆ K0,−n−1 = K

which deﬁnes the FY∗ ﬁltration of type [−n, 0] of K :

i FY∗ K := K0,i−1. (9) fys

The filtered complex (K,FY∗ ) yields the spectral sequence in cohomology associated with the flag Y∗ : pq p+q p+q E1 (K,FY∗ ) = H (Y,Kp,p−1) =⇒ H (Y,K) (10) ssclh with induced filtration (we write F instead of F 0) of type [−n, 0] :

F p Hj(Y,K) = Ker (Hj(Y,K) −→ Hj(Y , i∗ K)). (11) ssclh1 Y∗ p−1 p−1 Similarly, we obtain the spectral sequence in cohomology with compact supports associated with the ﬂag Y∗

pq p+q ! p+q cE1 (K,FY∗ ) = Hc (Y, (Ip,p−1)∗(Ip,p−1) K) =⇒ Hc (Y,K) (12) ssclhc

with induced ﬁltration of type [0, n]:

F p Hj(Y,K) = Im (Hj(Y, i! K) −→ Hj(Y,K)) (13) ssclh2 Y∗ c c −p c

! ∗ ev1 Remark 3.2 Since Verdier Duality exchanges I with I and I! with I∗, there is a natural isomorphism of spectral sequences:

∨ ∨ cEr(K,FY∗ ) ' (Er(K ),FY∗ ) .

The distinguished triangles

+ K−i−1,−n−1[−1] −→ K−i,−i−1 −→ K−i,−n−1 −→

b yield a complex in C (DY ), i.e. ﬁrst set

K−i := K−i,−i−1[−i], (14) ki

∂n ∂−n+1 ∂−1 ∂0 0 −→ K−n −→ K−n+1 −→ K−1 −→ K0 −→ K −→ 0. (15) kn

9 Note that each K−i is obtained by restriction to Y−i \Y−i−1, followed by extension by zero to Y and shift by [−i]. The result is supported on Y−i. Note also that this operation is not in general t−exact, i.e. it does not preserve perverse sheaves. b Let K ∈ C (DY ). Using the construction (15) we obtain a bounded double complex

j j j 2,b (Ki , di , ∂i ) ∈ C (DY ), i ≤ 0,

b with associated single complex S(K). Note that there is the natural map in C (DY )

a : S(K) −→ K (16) rc

∂j j 0 j j 0 j deﬁned by K0 −→ K and Ki<0 −→ K .

onep Example 3.3 Let P ∈ PY and suppose that Y∗ is cellular for P, i.e. that

r H (Y,Pi) = 0, ∀ r 6= i.

In this case, the spectral sequence Er(P,FY∗ ) is the spectral sequence of the complex 0 H (Y,P•): −n ∂−n ∂0 0 0 −→ H (Y,P−n) −→ ... −→ H (Y,P0) −→ 0 endowed with the bˆete ﬁltration σ :

p 0 p ∂p ∂0 0 σ H (Y,P•) := [ 0 −→ H (Y,Pp) −→ ... −→ H (Y,P0) −→ 0 ].

r r0 r 0 In particular, H (Y,P ) = E2 (P,Y∗) = H (H (Y,P•)). Note that the bêtefiltration on Hr(Y,P ) is the perverse filtration. This is of course in agreement with our main result.

3.4 The perverse spectral sequence sstpss Let K ∈ DY . Verdier’s formalism of spectral objects in the context of the category DY , endowed with the perverse t−structure, yields the perverse spectral sequence in cohomology

st p s p t s+t E2 (K, τ) = H (Y, H (K)) =⇒ H (Y,K), (17) a1

with abutment the ﬁltration F pτ

s j j p j F pτ H (Y,K) = Im(H (Y, τ≤−sK) −→ H (Y,K)). (18) a11

One deﬁnes L pτ via (2). There is the perverse spectral sequence for cohomology with compact supports

st p s p t s+t cE2 (K, τ) = Hc (Y, H (K)) =⇒ Hc (Y,K), (19) a2

with associated ﬁltrations F pτ and L pτ , e.g.:

s j j p j L pτ Hc (Y,K) = Im(Hc (Y, τ≤−s+jK) −→ Hc (Y,K)). (20) a22

10 vd2 Remark 3.4 Verdier Duality and the canonical isomorphisms pHi(K∨) ' pH−i(K)∨, yield a canonical isomorphism of spectral sequences

p ∨ p ∨ cEr(K, τ) ' (Er(K , τ)) . cnoc Remark 3.5 By virtue of Remarks 3.2 and 3.4, in the context of the perverse spectral sequence and of the spectral sequence of a ﬂag, the study of compactly supported cohomology is reduced to the study of cohomology. This contrast the case of the standard t−structure, i.e. of the Grothendieck and Leray spectral sequences. These spectral sequences are not well-behaved with respect to duality.

The main reason for considering the filtration L pτ is as follows: on one side, the interval p i type of F pτ depends on the interval for which H (K) 6= 0 and hence it varies with K; on the other hand, the interval type of L pτ depends only on the topology of Y. By Theorem 2.2 on the cohomological dimension of affine varieties, if Y is affine, then the type of L pτ is [−n, 0] for cohomology, and [0, n] for cohomology with compact

supports. Those are precisely the types for the ﬁltrations FY∗ associated with a n−ﬂag Y∗ of subspaces of Y.

p This suggest the possibility of a non trivial relation between L τ and FY∗ . By Arapura’s result [1], if Y is affine, then there is a flag of hyperplane sections for which one has the equality with the L−filtration associated with the standard Grothendieck filtration. This flag is made of special hyperplane sections of Y. Our main result is that if one chooses general hyperplane sections of the affine variety Y, then we have

p L τ = FY∗ .

3.5 The Jouanolou Trick ssjo The following is due to Jouanolou.

jo Proposition 3.6 Let Y be a quasi projective variety. There exists a Zariski locally trivial d A −ﬁbration π : Y → Y with Y an aﬃne variety.

Proof. See [8]. A proof that yields the sharp estimate d = n = dim Y goes as follows. Take a projective completion Y ⊆ Y. By blowing up the boundary, there is no loss of generality in assuming that the boundary is a Cartier divisor. Take any rank n + 1 vector bundle E on Y which is ample and generated by global sections. Let H ⊆ PY (E) be the zero set of a general section of the ample O(1). The map Y := P(E)Y \ H → Y is the wanted map for Y. The restriction to Y is the one for Y. imjo Remark 3.7 The fibration is not unique in any essential way. In [8], the integer d = d(Y ) is not effective. The bound d = dim Y is sharp with respect to the existence of the fibration. Compare this with Remark 5.3.

11 The following Lemma reduces the study of the perverse spectral sequence on a quasi projective variety, to the case of affine varieties. The notions of translated spectral sequence E1(l) and translated filtration L(l) are defined in section 3.1.

pjc Lemma 3.8 Let Y be a quasi projective variety, π : Y → Y be a ﬁbration as in Proposition 3.6 and K ∈ DY . There are canonical isomorphisms of spectral sequences

pq p pq ∗ p E2 (K, τ) 'E2 (π K, τ)(−d),

pq p pq ! p cE2 (K, τ) ' cE2 (π K, τ)(d) and canonical isomorphisms of ﬁltered Q−vector spaces.

∗ (H(Y,K),L pτ ) ' (H(Y, π K),L pτ (−d)),

! (Hc(Y, π K),L pτ (d)) ' (Hc(Y,K),L pτ ). Proof. The statements are simple consequences of the natural isomorphisms

∗ ! Id ' π∗π , π!π ' Id,

due to the contractibility of the ﬁbers of the bundle projection π, and to the t−exactness of π∗[d] ' π![−d], due to the fact that π is smooth of relative dimension d.

4 Resolutions in derived category of perverse sheaves sderper The goal of this section is to construct suitably acyclic resolutions of complexes K in the b derived category of perverse sheaves D (PY ). This is achieved in Proposition 4.9 by means of a ﬂag Y∗ of hyperplane sections adapted to the strata of a stratiﬁcation of K.

4.1 Beilinson Equivalence Theorem ssecbet ≤0 ≥0 A t−category D is a triangulated category endowed with a t−structure: D ,D , τ≤i, i τ≥i,H , the heart C, etc. We work with the following three t−categories: DY with the perverse t−structure, b b D (PY ) and D (VQ),VQ the category of Q−vector spaces, with their natural standard t−structure. A functor of t−categories T : D → D0 is a functor of the underlying triangulated categories, i.e. T is additive, commutes with translation and preserves distinguished triangles. The functor t is right t−exact (resp. left t−exact, resp. t−exact) if TD≤0 ⊆ D0≤0 (resp. TD≥0 ⊆ D0≥0, resp. T C ⊆ C0.)

12 beq Theorem 4.1 (Beilinson’s Equivalence) There is an equivalence of t−categories called the realization functor b ' rY : D (PY ) −→ DY ,

i.e. rY is a functor of t−categories which is t−exact and is an equivalence.

The functors rY and sY are constructed as follows, see [3] 3.1.7: Let DFY be the filtered derived category whose objects are finitely filtered complexes of sheaves with constructible cohomology, and ω : DFY → DY be the functor which forgets the filtration. The perverse t-structure on DY induces a t-structure on DFY . Its heart, DFβ is the abelian category of complexes K endowed with a bête filtration F, that is

i grF (K)[i] ∈ PY for all i. It is possible to deﬁne a functor

b H : DFβ → C (PY ) by

j j j H(K,F ) := P := grF (K)[j] with diﬀerentials dj : P j → P j+1 induced by the exact sequences

j+1 j j+2 j 0 → grF (K) → F /F (K) → grF (K) → 0. The following is proved in [3] Proposition 3.1.8:

b Proposition 4.2 The functor H : DFβ → C (PY ) is an equivalence of abelian categories.

−1 b The functor rY is then deﬁned from the composition ω ◦ H : C (PY ) → DY once b it is proved that quasi isomorphic complexes in C (PY ) are sent to isomorphic objects of DY . By the theorem of Beilinson, [2], rY is an equivalence of categories. For every complex 0 0 b b K ∈ DY , there exists (K ,F ) ∈ DFβ such that K ' K in DY . Let loc : C (PY ) → D (PY ) be the localization functor. One deﬁnes 0 sY (K) = loc(H(K ,F )).

This deﬁnes a quasi inverse to rY . we have the following:

H b DFβ / C (PY ) (21) diagrammo

ω loc sY - b DY k D (PY ) rY

13 4.2 Cohomology as a left derived functor cldf Let RΓ(Y, −) be the right derived functor of the global section functor Γ(Y, −). One has Hr(Y, −) = Hr ◦ RΓ(Y, −). There is the commutative diagram

b rY D (PY ) / DY (22) rzero 8 J 8 JJ 8 JJR 88 JJ RΓ 8 JJ 88 0 % 8R Db(V ) 88 Q 88 8 0 88 H 8 VQ

0 j j The functor R is cohomological and R (K) = H (Y, rY (K)). Let Y be aﬃne. The theorem on the cohomological dimension of aﬃne varieties 2.2 0 implies that R| : PY → VQ is right exact. b 0 j l 0 A complex K ∈ D (PY ) is R −acyclic if R (K ) = 0, for all j 6= 0. A R −acyclic left resolution of K is a quasi isomorphism (qis) L → K with L a R0−acyclic complex.

brt Theorem 4.3 (Cohomology as a left derived functor) Let Y be aﬃne. Then the b b 0 functor R : D (PY ) → D (VQ) is the left derived functor of the functor R| : PY → VQ :

0 R = L(R| ).

b 0 Every K ∈ D (PY ) admits a R −acyclic left resolution L → K and

0 • b R(K) = R (L ) in D (VQ),

j j j 0 • H (Y, rY (K)) = R (K) = H (R (L )).

Proof. See [2]. 0 0 If Y is not affine, then R| cannot be derived, due to the absence of enough R −acyclic objects. However, R can be computed using a Cech-type complex associated with a finite affine covering. See [2], section 3.

4.3 R0−acyclic filtered resolutions ssecrafr 0 b A R −acyclic filtered left resolution of a filtered complex (K,F ) in C (PY ) is a filtered map of complexes (L, G) → (K,F ) such that p p (1) the induced maps grGL → grF K are all qis and p 0 (2) the grGL are all R −acyclic. The finiteness of G implies that (2), (20) and (200) below are equivalent: (20) the GpL are all R0−acyclic;

14 (200) the GpL/GpL are all R0−acyclic. Using Verdier’s notion of spectral object [7], given the filtered complex (K,F ) one has a spectral sequence p+q p p+q E1(K,F ) := R (grF K) =⇒ R (K), with abutment F 0pRj(K) = Im(Rj(F pK) → Rj(K)). If (1) above one does not hold, then Lemma 4.4 does not hold. If (1) holds for (L, G), then 0 • one obtains the filtered complex of Q−vector spaces (R (L ), G, where G is the filtration defined by the injective images

R0(GpL•) −→ R0(L•).

This filtered complex has its own spectral sequence with abutment (Rj(L), G0). The following is a standard fact about acyclic resolutions which is proved by a simple and systematic use of condition (1). risoaci Lemma 4.4 Let (L, G) → (K,F ) be a R0−acyclic filtered left resolution. Then there are canonical identifications of spectral sequences and of their abutments:

0 • Er(R (L ), G) ' Er(L, G) ' Er(K,F ),

(Hj(R0(K•)), G0) ' (Rj(L),G0) ' (Rj(K),F 0).

4.4 Double complexes: the ﬁltrations F1 and F2 := Dec(F1) ssecdc Let A be an additive category. We are interested in the case A = DY . This is just a matter of exposition style used on the way to proving that the object of interest to us are in fact in the abelian category A = PY ; see Proposition 4.9. b i i i+1 A bounded complex (K, d) ∈ C (A) is a collection d : K → K , i ∈ Z of objects and maps in A such that Ki = 0 for |i| 0 and di+1 ◦ di = 0. 2,b j j j A bounded double complex (K, d, ∂) ∈ C (A) is a collection (Ki , dj, ∂i ), i, j ∈ Z, of objects and maps j+1 Ki O j di ∂j j i j Ki / Ki+1,

j 2 2 such that Ki = 0 for |i|, |j| 0, and d = 0, ∂ = 0 and d∂ = ∂d. The associated single complex functor

S : C2,b(A) −→ Cb(A), (K, d, ∂) 7−→ (S(K),D)

15 where j M j S (K) := Ki i+j=l and D is deﬁned componentwise as

l l−u u l−u D l−u := du + (−1) ∂u . |Ku Let A be abelian. The complex S(K) admits several natural decreasing filtrations. i (1) The standard filtration τ : τ S(K) := τ≤−iS(K). p • (2) The filtration by columns F1 : F1 S(K) is obtained by deleting the columns Kp0 , p0 < p and forming the associated single complex. p (3) The shifted filtration F2 := Dec(F1). It is immediate to verify that F2 (S(K)) is p0 a the single complex of a double complex, i.e. the one obtained by deleting the rows K• with p0 > −p, by keeping the rows with p0 < −p and by replacing the (−p)−th row with −p the row ker d• .

Note that while in the filtration by columns F1 one deletes “earlier” columns, the shifted filtration deletes later rows but keeps the foremost row of kernels. This is in analogy with the bête and standard filtrations of a complex.

4.5 Resolutions of perverse sheaves via hyperplane sections rpshs Let (K, d) ∈ DY . Given a n−ﬂag Y∗ on Y, the functoriality of the construction (15) yields 2,b a double complex in C (DY ) j j j (Ki , di , ∂i ) b Deﬁne a complex in C (DY ) by setting:

SY∗ (K) := S(K, d, δ). (23) sks

b There is the natural map (16) of complexes in C (DY ):

a : S(K) −→ K. (24) nca

By definition, a n−flag is a nested sequence of closed subvarieties. Given an ordered set of n closed subvarieties H1,...,Hn of Y, one gets a n−flag by setting Y0 := Y,Y−i := H1 ∩ ... ∩ Hi and Y−n−1 := ∅. Changing the order changes the flag. prep1 Lemma 4.5 Let Y be quasi projective of dimension n. For every P ∈ PY there is a n−flag b Y∗ such that the complex (15) in C (DY )

∂−n ∂−1 ∂0 0 −→ P−n −→ ... −→ P0 −→ P −→ 0 (25) pn

b coincides with SY∗ (P ), is in C (PY ), is exact in PY and is functorial in P.

16 Proof. What follows is a mere careful iteration of the procedure in the proof of Lemma 2.1. Fix an embedding Y ⊆ P. Let Σ be a stratiﬁcation for which P ∈ PY Σ. STEP 1. Choose a hyperplane section H1 according to Lemma 2.1, i.e. not containing any j1 i1 stratum. Let Y \ H1 → Y ← H1 be the obvious maps. We get the short exact sequences in PY : ∗ ! 0 −→ i1∗i1 P [−1] −→ j1!j1P −→ P −→ 0. ! Set P0 := j1!j1P. 1 Reﬁne Σ to Σ so that H1 becomes a union of strata. The exact sequence above takes Σ1 place in PY . ∗ Σ1 STEP 2. Repeat step one for i1∗i1 P [−1] ∈ PY , and get an exact sequence in PY : ∗ ∗ ! ∗ ∗ 0 −→ i2∗i2 (i1∗i1 P [−1]) −→ j2!j2i1∗i1 P [−1] −→ i1∗i1 P [−1] −→ 0,

j2 i2 where Y \ (H1 ∩ H2) → Y ← H1 ∩ H2. ! ∗ 1 2 Set P−1 := j2!j2i1∗i1 P [−1]. Reﬁne Σ to Σ so that H1 ∩ H2 and H2 become union of strata. Σ2 We obtain the exact sequence in PY ∗ ∗ 0 −→ i2∗i2 (i1∗i1 P [−1]) −→ P−1 −→ P0 −→ P −→ 0.

Repeat until we reach STEP n + 1 and the wanted exact sequence. Clearly, the P−i constructed here coincide with the ones of (15).

Σ genhypok Remark 4.6 Let P ∈ PY . The n−flag Y∗ can be chosen so that it depends only on Σ, i.e Σ so that the flag is such that the conclusions of Lemma 4.5 hold for every P ∈ PY . The flag constructed in Lemma 4.5 is the flag associated with the choice of n hyperplane sections Hi for n arbitrary embeddings: H1 has to meet the requirement of Lemma 2.1 for Σ; H2 1 should meet the requirement for the refinement for Σ of Σ that incorporates H1, and so on.

An inspection of the proof of Lemma 4.5 yields the following exactness statements

Σ prep2 Lemma 4.7 Let Σ be a ﬁxed stratiﬁcation of Y and P ∈ PY . Let Hi be as in the proof of Lemma 4.5. Then the functors

Σ (−)i : PY −→ PY ,P 7−→ Pi are exact for all −n ≤ i ≤ 0. In particular, the functor

Σ b PY −→ C (PY ),P 7−→ [0 → P−n → ... → P0 → 0] is exact.

Proof. Clearly, the ﬁrst statement implies the second. The ﬁrst statement is proved by applying Lemma 2.1 in each of the Steps of the proof of Lemma 4.5 and by using the exact

functors PYi → PY .

17 acycp Remark 4.8 If Y is aﬃne, then Theorem 2.3, coupled with Remark 4.6, allows to choose Σ the hyperplanes Hi so that (25) is an acyclic resolution of every P ∈ PY , with respect to 0 0 the functor R = H (Y, rY (−)) deﬁned earlier in (22):

j H (Y,Pi) = 0, ∀j 6= 0, ∀i.

b 4.6 Resolutions in C (PY ) using hyperplane sections ruhs Recall the content of section 4.3 on acyclic ﬁltered resolutions, and the single complex construction of section 4.4. b j Let (K, d) ∈ C (PY ). Resolving each entry K in the manner of Lemma 4.5 yields the b b map (24) of complexes in C (PY ), i.e. not just merely in C (DY ),

b a : SY∗ (K) −→ K, in C (PY ). (26) mapa

The explicit nature of the map (26) makes it clear that it induces maps of ﬁltered complexes: a00 a0 (SY∗ (K),F2) = Dec(F1)) −→ (K, τ) ←− (SY∗ (K), τ). (27) aaaof Note that a000 is ﬁltered just because a is a map of complexes.

b mpr Proposition 4.9 Let Y be affine of dimension n, K ∈ C (PY ). There is a flag Y∗ such b that for its associated map (26) of complexes in C (PY ) a : SY∗ (K) → K one has 0 (1) The map a :(SY∗ (K),F1) → K is a R −acyclic left resolution. 00 0 (2) The map a :(SY∗ (K),F2) → (K, τ) is a R −acyclic filtered left resolution. (3) There is the identity of complexes of vector spaces

0 • 0 • (R (SY∗ (K), Dec(F1)) = (R (SY∗ (K), F2)).

b Proof. Let Σ be a stratiﬁcation such that K ∈ C (PY ). Note that this implies that every j j j Ker d , Im d and H (K), etc. is in PY . By Remark 4.8, Lemma 4.5 yields a n−ﬂag Y∗ that simultaneously and functorially resolves the perverse sheaves Kj, Ker dj, Coker dj and j 0 H (K) in PY while at the same time ensures that the resolutions are R −acyclic. This Σ means that for all the perverse sheaves above there are the PY −functorial resolutions (25) 0 and that the entries Pi are R −acyclic. j (1). By Lemma 4.5, the columns (25) of the double complex Ki are exact. It is a standard fact that this implies that a0 is a qis. In the remainder of the proof, the exactness of the functors (−)i, established in Lemma 4.7, is essential. p p (2). The property to be proved can be tested against the F2 . The complex F2 SY∗ (K) is the single complex associated with the double complex described in section 4.4. Its l−th row is a R0−acyclic resolution of Kl, for l < −p, of Ker d−p for l = −p and is zero for p p 0 l > −p. As in the proof of (1), the map F2 SY∗ (K) → τ≤−pK =: τ K is a R −acyclic left resolution.

18 (3). We need to show that

j j 0 j 0 j di 0 j+1 0 j di j+1 0 j Ker R (di ) = Ker (R (Ki ) → R (Ki )) = R (Ker (Ki → Ki )) = R (Ker di ).

j There is the canonical factorization of d in PY

epi Kerdj mono−→ Kj −→ Imdj mono−→ Kj+1.

The application of the exact functor (−)i to the factorization above yields the factorization j of di : j mono j epi j mono j+1 Ker di −→ Ki −→ Im di −→ Ki . By the R0−acyclicity of all entries, we have

0 j mono 0 j epi 0 j mono 0 j+1 R (Ker di ) −→ R (Ki ) −→ R (Im di ) −→ R (Ki ) which is the desired conclusion.

fco Remark 4.10 The proof shows that the n−flag Y∗ can be chosen to be as in Remark 4.6 with the additional restrictions imposed by the acyclicity vanishings. These are conditions at infinity; see Remark 2.4. Summarizing: the n−flag Y∗ of Proposition 4.9 can be chosen to be the flag associated with n−general hyperplanes.

5 The geometry of the perverse ﬁltration smr In this section we state and prove the main result of this paper, Theorem 5.1. The proof is given in section 5.4, after some further preparation.

5.1 The statement of the main result somr Recall that translated spectral sequences and ﬁltrations are deﬁned in section 3.1.

mtm Theorem 5.1 (Geometry of the perverse ﬁltration) Let Y be a quasi projective variety of dimension n and K ∈ DY . Let π : Y → Y and d be as in Proposition 3.6. There exists a ﬂag Y∗ on Y such that (1) (cohomology) there is a natural isomorphism of spectral sequences

p ∗ Er(K, τ) ' Er(π K, Y∗)(−d), r ≥ 2

and ∗ p L τ = FY∗ (−d) via H(Y,K) ' H(Y, π K); (2) (compact supports) there is a natural isomorphism of spectral sequences

p ! cEr(K, τ) ' cEr(π K, Y∗)(d), r ≥ 2

19 and ! p L τ = FY∗ (d) via Hc(Y, π K) ' Hc(Y,K).

Moreover, if Y is aﬃne, then one may take π = IdY , d = 0 and, in particular,

p j j j L pτ H (Y,K) = Ker(H (Y,K) −→ H (Yp−1,K|Yp−1 )),

p j j ! j L pτ Hc (Y,K) = Im(Hc (Yp, ipK) −→ Hc (Y,K)).

pul Remark 5.2 Lemma 3.8, i.e. the use of the Jouanolou Trick, reduces the proof of this result to the case of Y affine. As the proof shows, one chooses the flag Y∗ by taking general hyperplane sections of Y. The filtrations can be viewed as kernels and images associated with the maps πl := π ◦ il : Yl → Y. If Y is affine, Y = Y and the flag is on Y.

loff Remark 5.3 The filtrations in cohomology are of type [−n, d] when Y is quasi projective, and of type [−n, 0] when Y is affine. The filtrations in cohomology with compact supports are of type [−d, n] if Y is quasi projective and of type [0, n] is Y is affine. On a given quasi projective variety, the bounds [−n, d] and [−d, n] are not sharp. Sharp bounds can be obtained using, instead of the Jouanolou trick, a Cech-type argument involving an affine covering with (1 + t) open sets and t minimum with this property. The sharp bounds are [−n, t] and [−t, n]. On the other hand, with this Cech-type approach, one looses the interpretation of the filtration in terms of kernels and images from actual varieties. They are kernels and images in the context of the formalism of simplicial schemes. This is enough for applications to mixed Hodge theory.

cnoc2 Remark 5.4 The case of cohomology with compact supports is reduced to the case of ordinary cohomology by virtue of Remark 3.5, i.e. by Verdier Duality. Of course, as an alternative, one may carbon-copy the proof in cohomology and make the obvious changes. 0 0 The main formal diﬀerence to keep in mind is that the functor Rc , i.e. the analogue of R for compact supports, is left exact and one ends up with right resolutions instead of the left resolutions of this paper.

5.2 Two lemmas on biﬁltered complexes bifico 2 Let D FY the bounded biﬁltered derived category of sheaves on Y . There is the diagonal 2 functor δ : D FY → DFY , see 3.2.

2 2 bbete Deﬁnition 5.5 The subcategory D Fβ ⊆ D F is deﬁned as follows:

2 i j D Fβ = {(K,F,G) such that grF grG(K)[i + j] ∈ PY }.

2 2 Just as in the case of DFβ ⊆ DF, there is a t-structure on D FY with heart D Fβ. 2 0 We note that the functor δ restricts to δ : D Fβ → DFβ. There is a functor H : 2 2,b D Fβ → C (PY ) to the category of bounded double complexes of perverse sheaves.

0 j j j i H (K,F,G)i := Pi := grF grG(K)[i + j]

20 j j+1 j j+1 with the diﬀerentials d : Pi → Pi , ∂ : Pi → Pi induced respectively by the exact sequences j+1 i j j+2 i j i 0 → grF grG(K) → F /F (grG(K)) → grF grG(K) → 0 and

i+1 j i i+2 j i j 0 → grG grF (K) → G /G (grF (K)) → grGgrF (K) → 0

2,b b total Lemma 5.6 Let S : C (PY ) → C (PY ) be the simple complex functor. The following diagram of functors 0 2 H 2,b D Fβ / C (PY )

δ S H b DFβ / C (PY ) is commutative.

Proof Without loss of generality we can suppose that the double ﬁltration is split in every degree. In this case the statement of the lemma can be veriﬁed by a direct computation.

2 i diago Remark 5.7 If (K,F,G) ∈ D Fβ then, the complexes G K, endowed with the restriction 2 of the filtrations F and G, still belong to D Fβ, since the condition 5.5 is tested for all i, j so that in particular it holds for all i0 ≤ i. i i j In particular, (G , δ(F,G)|Gi ) ∈ DFβ for every i. Similarly, all the quotients G K/G K, with the filtration induced by δ(F,G), belong to DFβ. We note that this is a property of the diagonal filtration which does not hold in general: 0 1 e.g. suppose K ∈ PY so that a bêtefiltration is F = K,F = {0}. Since the complex K0,−2 is not perverse in general, the restriction of F to K0,−2 may fail to be bête.

l Suppose, to ﬁx the notation, that grG(K) = 0 for l∈ / [−n, 0]. We may now consider the following ﬁltered object in DFβ : (in order to keep the notation reasonable we will denote δ(F,G) simply by δ and H(K,F ) by H(K). )

0 −1 −n+1 −n (G , δ|G0 ) ⊆ (G , δ|G−1 ) ⊆ · · · ⊆ (G , δ|G−n+1 ) ⊆ (G , δ|G−n ) = (K, δ)

b Applying the equivalence of abelian categories H we obtain a ﬁltered object in C (PY ):

H(G0) ⊆ H(G−1) ⊆ · · · ⊆ H(G−n+1) ⊆ H(G−n) = H(K).

Recall the deﬁnition of the ﬁltration F1 from section 4.4: We have the following lemma, whose proof is immediate:

21 2 compa Lemma 5.8 Let (K,F,G) ∈ D Fβ. Then there is a natural identiﬁcation of ﬁltered objects

⊆ ⊆ ⊆ ⊆ H(G0) / H(G−1) / ··· / H(G−n+1) / H(G−n) = H(K)

= = = = 0 ⊆ −1 ⊆ ⊆ −n+1 ⊆ −n F1 / F1 / ··· / F1 / F1 = H(K)

5.3 Construction of an isomorphism of spectral sequences constru Let K ∈ DY be a complex of injective sheaves, and Y∗ : ∅ ⊆ Y−n ⊆ · · · ⊆ Y−1 ⊆ Y0 = Y a sequence of closed subschemes of Y . There is the chain of inclusions

0 ⊆ K0,−1 ⊆ ... ⊆ K0,−n ⊆ K0,−n−1 = K

which deﬁnes the FY∗ ﬁltration of type [−n, 0] of K. The complexes K0,i are still Γ−acyclic. The spectral sequence

pq p+q p+q E1 (K,FY∗ ) = H (Y,Kp,p−1) =⇒ H (Y,K) arises from the ﬁltered complex of abelian groups

{0} ⊆ Γ(Y,K0,−1) ⊆ · · · Γ(Y,K0,−n) ⊆ Γ(Y,K)

We can suppose, up to isomorphism in DY , that K is endowed with a bˆeteﬁltration F , which we may furthermore suppose split in any degree. We then have the complex of perverse sheaves sY (K) = H(K,F ), see Section 4.1.

bibete Lemma 5.9 Let Y be aﬃne and (K,F ) ∈ DFβ as above. Then there is an n−ﬂag Y∗ such that

SY∗ (sY (K)) = H(K, δ(F,FY∗ )) and

H(K, δ(F,FY∗ )) → H(K,F ) is a R0− acyclic resolution. Similarly:

i+1 i S (s (K0,i)) = H(F K, δ ), S (s (Ki,i−1)) = H(gr K, δ ). Y∗ Y Y∗ | Y∗ Y FY∗ |

i+1 i 0 The complexes H(F K, δ ) and H(gr L, δ ) are R −acyclic left resolutions of sY (K0,i) Y∗ | FY∗ | and sY (Ki,i−1), respectively. Finally, there are natural identiﬁcations

grp (S (s (K))) ' s (grp (K)). (28) elakey F1 Y∗ Y Y Y∗

22 i Proof. Since, by assumption, grF (K)[i] ∈ PY , by Lemma 4.7 we have

j i i gr gr (K)[i + j] = (gr (K)[i])j,j−1[j] ∈ PY . FY∗ F F

2 In particular (K,F,G) ∈ D Fβ and we conclude by 5.6. In view of Remark 5.7, applied to

the case G := FY∗ , the other complexes, and the last statement, are dealt with similarly.

We now compare the spectral sequences. Recall that the F ﬁltrations are induced by F via R0 and that the acyclicity conditions of Proposition 4.9 ensure that they are deﬁned as injective images. coss Proposition 5.10 There is a natural isomorphism of spectral sequences

pq 0 pq Er (R (SY∗ (sY (K))), F1) ' Er (K,Y∗), r ≥ 1.

Proof. The complex

SY∗ (sY (K0,−1)) ⊆ · · · ⊆ SY∗ (sY (K0,−n)) ⊆ SY∗ (sY (K)) ' sY (K)

is ﬁltered R0−acyclic and, in view of Lemma 5.9, is identiﬁed with

0 −n F1 SY∗ (sY (K)) ⊆ · · · ⊆ F1 SY∗ (sY (K)) ⊆ SY∗ (sY (K)) ' sY (K).

Recalling that R(C) = H(Y, rY (C)), the identiﬁcations (28) and that Beilinson’s Theorem

4.1 yields a natural isomorphism of functors rY ◦ sY ' IdDY , we have the following chain of natural isomorphisms of spectral sequences and of their abutments

p Hp+q(Y, gr K) Hp+q(Y,K) FY∗ +3 O O ' T h.4.1 '

p+q p p+q H (Y, rY sY gr K) H (Y, r s K) FY∗ +3 Y Y O O = =

p+q p p+q R (sY gr K) R (s K) FY∗ +3 Y O O ' (28) '

Rp+q(grp S s K) p+q F1 Y∗ Y +3 R (SY∗ sY K).

23 5.4 Proof of Theorem 5.1 pmtmm Proof. By Remarks 5.2 and 5.4, we are reduced to proving the result for cohomology. on an affine Y. Keeping in mind the distinct notions of re-numeration and of shift of filtration with its effect on the spectral sequence, we have the following chain of natural isomorphisms of spectral sequences for r ≥ 1 :

E (K,Y ) P r. 5.10 E (R0(S• (s (K)), F ) r+1 ∗ ' r+1 Y∗ Y 1

(7) 0 • Er+1(R (S (s (K)), Dec(F1)) ' Y∗ Y

P r. 4.9.3 E (R0(S• (s (K)), F ) ' r+1 Y∗ Y 2

P r. 4.9.2 p ' Er+1(K, τ). 0 0 p The ﬁltrations F1 and F2 are identycal, modulo the shift (5). It follows that L τ = FY∗ on H(K).

6 Some Hodge-theoretic applications shta We state the result below in the case when Y is affine. Clearly, there is an analgous statement for Y quasi projective via the use of a fibration π : Y → Y as in Proposition 3.6 and the use of Lemma 3.8. flplr Theorem 6.1 Let f : X → Y be a map of varieties with Y affine. and C ∈ DX . Then there is a flag X∗ on X such that there is a natrual isomorphism of spectral sequences

p Er+1(C,X∗) 'Er+1(f∗C, τ), r ≥ 1.

Proof. Choose the hyperplane sections Hi in the proof of our main Theorem 5.1 so that, in addition to the exactness and acyclicity properties, one has the base change property for C in the Cartesian squares Xi / X

Yi / Y. This is made possible, for example, by using the Generic Base Change Theorem. It follows that Theorem 5.1 identiﬁes the perverse Leray spectral sequence for C, i.e. the perverse −1 spectral sequence for f∗C, with the spectral sequences for the ﬂag X∗ := f (Y∗) for the complex C.

24 mhspl Corollary 6.2 Let f : X → Y be a map of varieties with Y quasi projective. Then the perverse spectral sequences and resulting ﬁltrations

s p t s+t s p t s+t H (Y, H (f∗QY )) =⇒ H (X, Q),Hc (Y, H (f!QY )) =⇒ Hc (X, Q)

are of MHS.

Proof. We apply the quasi projective version of Theorem 6.1 to the special case C = QX . By standard mixed Hodge theory on algebraic varieties, the spectral sequences E(QX ,X∗) are all of mixed Hodge structures, regardless of the (algebraic) ﬂag.

The results that we state below extend to the case of a quasi projective target, the results of our paper [6], proved for a projective target. Using the reduction techniques in [5], [6], the proofs require a systematic application of of Corollary 6.2, for both H(X, Q) and Hc(X, Q), when X is smooth. The details of the proofs will appear elsewhere. ihmhs Theorem 6.3 Let Y be a quasi projective variety. Then the intersection cohomology groups IH(Y ) and IHc(Y ) admit natural MHS compatible with resolution of singularities. The Poincar´epairing isomorphism is of MHS. The natural maps H(Y ) → IH(Y ),Hc(Y ) → IHc(Y ) are of MHS. If Y is complete, then j j the kernel is Wj−1H (Y, Q) ⊆ H (Y, Q) (weights ≤ j − 1).

Let f : X → Y be a proper map with Y quasi projective. The Decomposition Theorem asserts the following: there are non canonical isomorphisms

j M j−dim X p i M M j−dim X−i φ : IH (X, Q) ' H H (f∗QX )[−i] = IH (Ya,La), (29) dt i i a∈Ii

where each Ya is an irreducible closed subvariety of Y,La is a semisimple local system on a smooth open subvariety of Ya and for every ﬁxed perverse index i, the varieties Ya, a ∈ Ii are distinct. As a short hand we write M M φ : IH(X) ' Ha i a∈Ii dtmhs Theorem 6.4 Each summand Ha inherits a natural MHS as a subquotient of the MHS on IH(X). Ditto for IHc.

eta Theorem 6.5 Let f : X → Y be projective of quasi projective varieties. Then there exist splitting φ that are isomorphisms of MHS.

25 sato Remark 6.6 The identification of the flag and perverse Leray spectral sequence in Theo- rem 6.1 is new. The mixed Hodge-theoretic nature of the perverse Leray spectral sequence is due to M. Saito, who used mixed Hodge modules. Theorems 6.3 and 6.4 are due to M. Saito. We can prove that the mixed Hodge structures arising in M. Saito’s results, and which stem from the general theory of mixed Hodge modules, coincide with the ones we find stemming from the mixed Hodge structure for relative cohomology, i.e. from the flag spectral sequences.

References

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Authors’ addresses: Mark Andrea A. de Cataldo, Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA. e-mail: [email protected]

26 Luca Migliorini, Dipartimento di Matematica, Universit`adi Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, ITALY. e-mail: [email protected]