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The Perverse Filtration and the Lefschetz Hyperplane Theorem

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The Perverse Filtration and the Lefschetz Hyperplane Theorem 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 Poincar´epairing 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 equiv- alent to the bounded-derived-type category of complexes with constructible cohomology. This equivalence transforms the standard truncation in the former category into the per- verse 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.
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