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The Decomposition Theorem, Perverse Sheaves and the Topology of Algebraic Maps

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The Decomposition Theorem, Perverse Sheaves and the Topology of Algebraic Maps The Decomposition Theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo∗ and Luca Miglioriniy July, 2008 Abstract We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples. Contents 1 Overview 3 1.1 The topology of complex projective manifolds: Lefschetz and Hodge Theorems 3 1.2 Singular algebraic varieties. 4 1.3 Families of smooth projective varieties . 6 1.4 Provisional statement of the Decomposition Theorem. 7 1.5 Crash Course on the derived category . 7 1.6 Decomposition, Semisimplicity and Relative Hard Lefschetz Theorems . 11 1.7 Invariant Cycle Theorems . 12 1.8 A few examples . 13 2 Applications of the Decomposition Theorem 14 2.1 Toric varieties and combinatorics of polytopes. 14 2.2 Semismall maps. 19 2.2.1 Semismall maps: strata and local systems . 23 2.2.2 Semismall maps: the semisimplicity of the endomorphism algebra . 24 2.3 The Hilbert Scheme of points on a surface. 25 ∗Partially supported by N.S.F. yPartially supported by GNSAGA 1 2.4 The Nilpotent Cone and Springer Theory. 27 2.5 The functions-sheaves dictionary and geometrization. 30 2.6 Schubert varieties and Kazhdan-Lusztig polynomials. 31 2.7 The Geometric Satake isomorphism. 38 2.8 Decomposition up to homological cobordism and signature . 44 3 Perverse sheaves 44 3.1 Intersection cohomology . 45 3.2 The Hodge-Lefschetz theorems for intersection cohomology . 45 3.3 Examples of intersection cohomology . 46 3.4 Definition and first properties of perverse sheaves . 48 3.5 The perverse filtration . 51 3.6 Perverse cohomology . 51 3.7 t-exactness and the Lefschetz Hyperplane Theorem . 53 3.8 Intermediate extensions . 54 3.9 Nearby and vanishing cycle functors . 56 3.10 Unipotent nearby and vanishing cycle functors . 58 3.11 Two descriptions of the category of perverse sheaves . 60 3.11.1 The approach of MacPherson-Vilonen . 60 3.11.2 The approach of Beilinson and Verdier. 63 4 Three approaches to the Decomposition Theorem 65 4.1 The proof of Beilinson, Bernstein, Deligne and Gabber . 65 4.1.1 Constructible Ql-sheaves . 66 4.1.2 Weights . 68 4.1.3 The structure of pure complexes . 70 4.1.4 The Decomposition Theorem over F . 72 4.1.5 The Decomposition Theorem for complex varieties . 73 4.1.6 From F to C ................................ 74 4.2 M. Saito's approach via mixed Hodge modules . 77 4.3 A proof via classical Hodge theory . 80 4.3.1 The results . 80 4.3.2 An outline of the proof. 82 5 Appendices 87 5.1 Hard Lefschetz and mixed Hodge structures . 87 5.2 The formalism in DY ............................... 89 5.2.1 Familiar objects from algebraic topology . 92 5.2.2 A formulary . 95 2 1 Overview The subject of perverse sheaves and the Decomposition Theorem have been at the heart of a revolution which took place over the last thirty years in algebra, representation theory and algebraic geometry. The Decomposition Theorem is a powerful tool to investigate the homological proper- ties of proper maps between algebraic varieties. It is the deepest fact known concerning the homology, Hodge theory and arithmetic properties of algebraic varieties. Since its discovery in the early 1980's, it has been applied in a wide variety of contexts, ranging from algebraic geometry to representation theory to combinatorics. It was conjectured by Gelfand and MacPherson in [78] in connection with the development of Intersection Homology theory. The Decomposition Theorem was then proved in very short order by Beilinson, Bernstein, Deligne and Gabber in [8]. In the sequel of this introduction we try to motivate the statement as a natural outgrowth of the deep investigations on the topological properties of algebraic varieties, begun with Riemann, Picard, Poincar´eand Lefschetz, and culminated in the spectacular results obtained with the development of Hodge Theory and ´etalecohomology. This forces us to avoid many crucial technical de- tails, some of which are dealt with more completely in the following sections. We have no pretense of historical completeness. For an account of the relevant history, see the historical remarks in [82], and the survey by Kleiman [111]. As to the contents of this survey, we refer to the detailed table of contents. Acknowledgments. We thank Pierre Deligne, Mark Goresky, Tom Haines, Andrea Maffei and Laurentiu Maxim for many conversations. We thank I.A.S. Princeton for the great working conditions in the A.Y. 2006/2007. During the preparation of this paper the second-named author has also been guest of the following institutions: I.C.T.P. in Trieste, Centro di Ricerca Matematica E. de Giorgi in Pisa. 1.1 The topology of complex projective manifolds: Lefschetz and Hodge Theorems Complex algebraic varieties provided an important motivation for the development of algebraic topology from its earliest days (Riemann, Picard Poincar´e).However, algebraic varieties and algebraic maps enjoy many truly remarkable topological properties that are not shared by other classes of spaces. These special features were first exploited by Lefschetz ([115]) (who claimed to have \planted the harpoon of algebraic topology into the body of the whale of algebraic geometry" [116], pag.13.) and they are almost completely summed up in the statement of the Decomposition Theorem and its embellishments. Example 1.1 (Formality) The relation between the cohomology and homotopy groups of a topological manifold is extremely complicated. Deligne, Griffiths, Morgan and Sullivan [60] have discovered that for a nonsingular simply connected complex projective variety X; the real homotopy groups πi(X)⊗R can be computed formally in terms of the cohomology ring H∗(X): 3 In the next few paragraphs we shall discuss the Lefschetz theorems and the Hodge decomposition and index theorems. Together with Deligne's degeneration theorem These are precursors of the Decomposition Theorem and they are essential tools in the proofs, described in x4.1,4.2,4.3, of the Decomposition Theorem. Let X be a complex smooth n-dimensional projective variety and let D = H \ X be the intersection of X with a generic hyperplane H. We use cohomology with rational coefficients. A standard textbook reference for what follows is [88]; see also [162] and [40]. The Lefschetz Hyperplane Theorem states that the restriction map Hi(X) ! Hi(D) is an isomorphism for i < n − 1 and is injective for i = n − 1: The cup product with the first Chern class of the hyperplane bundle gives a mapping i i+2 i \c1(H): H (X) ! H (X) which can be identified with the composition H (X) ! Hi(D) ! Hi+2(X), the latter being a \Gysin" homomorphism. The Hard Lefschetz Theorem states that for all 0 ≤ i ≤ n the i-fold iteration is an isomorphism i n−i ' n+i (\c1(H)) : H (X) −! H (X): Lefschetz had \proofs" of both theorems (see [112] for an interesting discussion of Lefschetz's proofs). R. Thom outlined a Morse-theoretic approach to the hyperplane theorem which was worked out in detail by Andreotti and Frankel (see [132]) [1] and Bott [18]. Lefschetz's proof of the Hard Lefschetz Theorem is incorrect. Hodge [95] (see also [165] gave the first complete proof. Deligne [58] gave a proof and a vast generalization of the Hard Lefschetz Theorem using varieties defined over finite fields. i p;q The Hodge Decomposition is the canonical decomposition H (X; C) = ⊕p+q=iH (X). The summand Hp;q(X) consists of cohomology classes on X which admit a g-harmonic (p; q) differential form as a representative, where g is any fixed K¨ahlermetric on X. For a fixed index 0 ≤ i ≤ n, there is the bilinear form on Hn−i(X) defined by Z i (a; b) 7−! (c1(H)) ^ a ^ b: X While the Hard Lefschetz Theorem is equivalent to the non degeneracy of these forms, the Hodge-Riemann Bilinear Relations (cf. x5.1, Theorem 5.1.(3) establish their signature properties. 1.2 Singular algebraic varieties. The Lefschetz and Hodge theorems fail if X is singular. Two (somewhat complementary) approaches to modifying them for singular varieties involve Mixed Hodge Theory [55, 56] and Intersection Cohomology [83, 84], [16]. In Mixed Hodge Theory the topological invariant studied is the same investigated for nonsingular varieties, namely, the cohomology groups of the variety, whereas the structure with which it is endowed changes. See [66] for an elementary and nice introduction. The (p; q) decomposition of classical Hodge Theory is replaced by a more complicated structure which is, roughly speaking, a finite iterated extension of (p; q)-decompositions. 4 The cohomology groups Hi(X) are endowed with an increasing filtration (the weight filtration) W; and the graded pieces Wk=Wk−1 have a (p; q) decomposition of weight k, that is p + q = k: Such a structure, called a mixed Hodge structure, exists canonically on any algebraic variety and satisfies several fundamental weight restrictions. Here are few: 1. if the not necessarily compact X is nonsingular, then the weight filtration on Hk(X) k starts at Wk; that is WrH (X) = 0 for r < k; 2. if X is compact and possibly singular, then the weight filtration on Hk(X) ends at k k k Wk; that is WrH (X) = WkH (X) = H (X) for r ≥ k: ∗ 1 1 Example 1.2 Let X = C : Then H (X) ' Q has weight 2.
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