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The Decomposition Theorem, Perverse Sheaves and the Topology Of The decomposition theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo and Luca Migliorini∗ 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 4 1.2 Families of smooth projective varieties . ........ 5 1.3 Singular algebraic varieties . ..... 7 1.4 Decomposition and hard Lefschetz in intersection cohomology . 8 1.5 Crash course on sheaves and derived categories . ........ 9 1.6 Decomposition, semisimplicity and relative hard Lefschetz theorems . 13 1.7 InvariantCycletheorems . 15 1.8 Afewexamples.................................. 16 1.9 The decomposition theorem and mixed Hodge structures . ......... 17 1.10 Historicalandotherremarks . 18 arXiv:0712.0349v2 [math.AG] 16 Apr 2009 2 Perverse sheaves 20 2.1 Intersection cohomology . 21 2.2 Examples of intersection cohomology . ...... 22 2.3 Definition and first properties of perverse sheaves . .......... 24 2.4 Theperversefiltration . .. .. .. .. .. .. .. 28 2.5 Perversecohomology .............................. 28 2.6 t-exactness and the Lefschetz hyperplane theorem . ...... 30 2.7 Intermediateextensions . 31 ∗Partially supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie” 1 3 Three approaches to the decomposition theorem 33 3.1 The proof of Beilinson, Bernstein, Deligne and Gabber . .......... 33 3.1.1 Constructible Ql-sheaves......................... 34 3.1.2 Weightsandpurity............................ 35 3.1.3 Thestructureofpurecomplexes . 37 3.1.4 The decomposition over F ........................ 38 3.1.5 The decomposition theorem for complex varieties . ....... 39 3.2 M. Saito’s approach via mixed Hodge modules . ..... 40 3.3 AproofviaclassicalHodgetheory . 43 3.3.1 The results when X is projective and nonsingular . 44 3.3.2 An outline of the proof of Theorem 3.3.1 . 46 4 Applications of perverse sheaves and of the decomposition theorem 50 4.1 Toric varieties and combinatorics of polytopes . .......... 50 4.1.1 The h-polynomial ............................ 51 4.1.2 Two worked out examples of toric resolutions . ..... 53 4.2 Semismallmaps.................................. 55 4.2.1 Semismall maps and intersection forms . 58 4.2.2 Examples of semismall maps I: Springer theory . ..... 59 4.2.3 Examples of semismall maps II: Hilbert schemes of points ...... 62 4.3 The functions-sheaves dictionary and geometrization . ............ 65 4.4 Schubert varieties and Kazhdan-Lusztig polynomials . ........... 66 4.5 TheGeometricSatakeisomorphism. 70 4.6 Ngˆo’ssupporttheorem. 75 4.7 Decomposition up to homological cobordism and signature.......... 78 4.8 Further developments and applications . ....... 78 5 Appendices 80 5.1 Algebraicvarieties .............................. 80 5.2 Hard Lefschetz and mixed Hodge structures . ...... 81 5.3 The formalism of the constructible derived category . .......... 83 5.4 Familiar objects from algebraic topology . ........ 89 5.5 Nearby and vanishing cycle functors . ..... 91 5.6 Unipotent nearby and vanishing cycle functors . ........ 94 5.7 Two descriptions of the category of perverse sheaves . .......... 95 5.7.1 The approach of MacPherson-Vilonen . 95 5.7.2 The approach of Beilinson and Verdier . 99 5.8 A formulary for the constructible derived category . ..........101 2 1 Overview The theory of perverse sheaves and one of its crowning achievements, the decomposition theorem, are at the heart of a revolution which has taken place over the last thirty years in algebra, representation theory and algebraic geometry. The decomposition theorem is a powerful tool for investigating the topological prop- erties of proper maps between algebraic varieties and is the deepest known fact relating their homological, Hodge-theoretic and arithmetic properties. In this 1, we try to motivate the statement of this theorem as a natural outgrowth of § the investigations on the topological properties of algebraic varieties begun with Lefschetz and culminated in the spectacular results obtained with the development of Hodge theory and ´etale cohomology. We gloss over many crucial technical details in favor of rendering a more panoramic picture; the appendices in 5 offer a partial remedy to these omissions. § We state the classical Lefschetz and Hodge theorems for projective manifolds in 1.1 and § Deligne’s results on families of projective manifolds in 1.2. In 1.3, we briefly discuss § § singular varieties and the appearance and role of mixed Hodge structures and intersec- tion cohomology. In 1.4, we state the decomposition theorem in terms of intersection § cohomology without any reference to perverse sheaves. The known proofs, however, use in an essential way the theory of perverse sheaves which, in turn, is deeply rooted in the formalism of sheaves and derived categories. We offer a “crash course” on sheaves in 1.5. § With these notions and ideas in hand, in 1.6 we state the decomposition theorem in terms § of intersection complexes (rather than in terms of intersection cohomology groups). We also state two important related results: the relative hard Lefschetz and semisimplicity theorems. 1.7 reviews the generalization to singular maps of the now classical properties § of the monodromy representation in cohomology for a family of projective manifolds. 1.8 § discusses surface and threefold examples of the statement of the decomposition theorem. 1.9 overviews the mixed Hodge structures arising from the decomposition theorem. We § provide a timeline for the main results mentioned in this overview in 1.10. § We have tried, and have surely failed in some ways, to write this survey so that most of it can be read by non experts and so that each chapter can be read independently of the others. For example, a reader interested in the decomposition theorem and in its applications could read 1, the first half of 4 and skim through the second half on § § geometrization, while a reader interested in the proofs could read 1 and 3. Perhaps, at § § that point, the reader may be motivated to read more about perverse sheaves. 2 is an introduction to perverse sheaves. In this survey, we deal only with middle § perversity, i.e. with a special case of perverse sheaves. It seemed natural to us to start this section with a discussion of intersection cohomology. In 2.3, we define perverse § sheaves, discuss their first properties, as well as their natural categorical framework, i.e. t-structures. In 2.4, we introduce the perverse filtration in cohomology and its geometric § description via the Lefschetz hyperplane theorem. 2.5 reviews the basic properties of the § cohomology functors associated with the perverse t-structure. 2.6 is about the Lefschetz § hyperplane theorem for intersection cohomology. In 2.7, we review the properties of the § 3 intermediate extension functor, of which intersection complexes are a key example. In 3, we discuss the three known approaches to the decomposition theorem: the orig- § inal one, due to A. Beilinson, J. Bernstein, P. Deligne and O. Gabber, via the arithmetic properties of varieties over finite fields, the one of M. Saito, via mixed Hodge modules, and ours, via classical Hodge theory. Each approach highlights different aspects of this important theorem. 4 contains a sampling of applications of the theory of perverse sheaves and, in par- § ticular, of the decomposition theorem. The applications range from algebraic geometry to representation theory and to combinatorics. While the first half of 4, on toric and on § semismall maps, is targeted to a general audience, the second half, on the geometrization of Hecke algebras and of the Satake isomorphism, is technically more demanding. Due to the fact that the recent and exciting development [152] in the Langlands program makes use of a result that deals with the decomposition theorem with “large fibers,” we have included a brief discussion of B.C. Ngˆo’s support theorem in 4.6. § The appendix 5 contains a brief definition of quasi projective varieties ( 5.1), of § § pure and mixed Hodge structures, the statement of the hard Lefschetz theorem and of the Hodge-Riemann relations ( 5.2), a description of the formalism of derived categories § ( 5.3), a discussion of how the more classical objects in algebraic topology relate to this § formalism ( 5.4), a discussion of the nearby and vanishing cycle functors ( 5.5), as well as § § their unipotent counterparts ( 5.6), two descriptions of the category of perverse sheaves § ( 5.7) and, finally, a formulary for the derived category ( 5.8). § § Unless otherwise stated, a variety is an irreducible complex algebraic variety and a map is a map of varieties. We work with sheaves of rational vector spaces, so that the cohomology groups are rational vector spaces. 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. On the other hand, algebraic varieties and algebraic maps enjoy many truly remarkable topological properties that are not shared by other classes of spaces and maps. These special features were first exploited by Lefschetz ([123]) (who claimed to have “planted the harpoon of algebraic topology into the body of the whale of algebraic geometry” [124], p.13) and they are almost completely summed up in the statement of the decomposition theorem and of its embellishments.
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