How to Glue Perverse Sheaves”
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Report for the Academic Year 1995
Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1994 - 95 PRINCETON NEW JERSEY Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1 994 - 95 OLDEN LANE PRINCETON • NEW JERSEY 08540-0631 609-734-8000 609-924-8399 (Fax) Extract from the letter addressed by the Founders to the Institute's Trustees, dated June 6, 1930. Newark, New jersey. It is fundamental in our purpose, and our express desire, that in the appointments to the staff and faculty, as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. TABLE OF CONTENTS 4 BACKGROUND AND PURPOSE 5 • FOUNDERS, TRUSTEES AND OFFICERS OF THE BOARD AND OF THE CORPORATION 8 • ADMINISTRATION 11 REPORT OF THE CHAIRMAN 15 REPORT OF THE DIRECTOR 23 • ACKNOWLEDGMENTS 27 • REPORT OF THE SCHOOL OF HISTORICAL STUDIES ACADEMIC ACTIVITIES MEMBERS, VISITORS AND RESEARCH STAFF 36 • REPORT OF THE SCHOOL OF MATHEMATICS ACADEMIC ACTIVITIES MEMBERS AND VISITORS 42 • REPORT OF THE SCHOOL OF NATURAL SCIENCES ACADEMIC ACTIVITIES MEMBERS AND VISITORS 50 • REPORT OF THE SCHOOL OF SOCIAL SCIENCE ACADEMIC ACTIVITIES MEMBERS, VISITORS AND RESEARCH STAFF 55 • REPORT OF THE INSTITUTE LIBRARIES 57 • RECORD OF INSTITUTE EVENTS IN THE ACADEMIC YEAR 1994-95 85 • INDEPENDENT AUDITORS' REPORT INSTITUTE FOR ADVANCED STUDY: BACKGROUND AND PURPOSE The Institute for Advanced Study is an independent, nonprofit institution devoted to the encouragement of learning and scholarship. -
Algebraic D-Modules and Representation Theory Of
Contemporary Mathematics Volume 154, 1993 Algebraic -modules and Representation TheoryDof Semisimple Lie Groups Dragan Miliˇci´c Abstract. This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple Lie groups. In particular, we discuss the theory of “localization” of modules over the envelop- ing algebra of a semisimple Lie algebra due to Alexander Beilinson and Joseph Bernstein [1], [2], and the work of Henryk Hecht, Wilfried Schmid, Joseph A. Wolf and the author on the localization of Harish-Chandra modules [7], [8], [13], [17], [18]. These results can be viewed as a vast generalization of the classical theorem of Armand Borel and Andr´e Weil on geometric realiza- tion of irreducible finite-dimensional representations of compact semisimple Lie groups [3]. 1. Introduction Let G0 be a connected semisimple Lie group with finite center. Fix a maximal compact subgroup K0 of G0. Let g be the complexified Lie algebra of G0 and k its subalgebra which is the complexified Lie algebra of K0. Denote by σ the corresponding Cartan involution, i.e., σ is the involution of g such that k is the set of its fixed points. Let K be the complexification of K0. The group K has a natural structure of a complex reductive algebraic group. Let π be an admissible representation of G0 of finite length. Then, the submod- ule V of all K0-finite vectors in this representation is a finitely generated module over the enveloping algebra (g) of g, and also a direct sum of finite-dimensional U irreducible representations of K0. -
Perverse Sheaves in Algebraic Geometry
Perverse sheaves in Algebraic Geometry Mark de Cataldo Lecture notes by Tony Feng Contents 1 The Decomposition Theorem 2 1.1 The smooth case . .2 1.2 Generalization to singular maps . .3 1.3 The derived category . .4 2 Perverse Sheaves 6 2.1 The constructible category . .6 2.2 Perverse sheaves . .7 2.3 Proof of Lefschetz Hyperplane Theorem . .8 2.4 Perverse t-structure . .9 2.5 Intersection cohomology . 10 3 Semi-small Maps 12 3.1 Examples and properties . 12 3.2 Lefschetz Hyperplane and Hard Lefschetz . 13 3.3 Decomposition Theorem . 16 4 The Decomposition Theorem 19 4.1 Symmetries from Poincaré-Verdier Duality . 19 4.2 Hard Lefschetz . 20 4.3 An Important Theorem . 22 5 The Perverse (Leray) Filtration 25 5.1 The classical Leray filtration. 25 5.2 The perverse filtration . 25 5.3 Hodge-theoretic implications . 28 1 1 THE DECOMPOSITION THEOREM 1 The Decomposition Theorem Perhaps the most successful application of perverse sheaves, and the motivation for their introduction, is the Decomposition Theorem. That is the subject of this section. The decomposition theorem is a generalization of a 1968 theorem of Deligne’s, from a smooth projective morphism to an arbitrary proper morphism. 1.1 The smooth case Let X = Y × F. Here and throughout, we use Q-coefficients in all our cohomology theories. Theorem 1.1 (Künneth formula). We have an isomorphism M H•(X) H•−q(Y) ⊗ Hq(F): q≥0 In particular, this implies that the pullback map H•(X) H•(F) is surjective, which is already rare for fibrations that are not products. -
IMU-Net 88: March 2018 a Bimonthly Email Newsletter from the International Mathematical Union Editor: Martin Raussen, Aalborg University, Denmark
IMU-Net 88: March 2018 A Bimonthly Email Newsletter from the International Mathematical Union Editor: Martin Raussen, Aalborg University, Denmark CONTENTS 1. Editorial: ICM 2018 2. IMU General Assembly meeting in São Paulo 3. CDC Panel Discussion and Poster Session during ICM 2018 4. Fields medalist Alan Baker passed away 5. CWM: Faces of women in mathematics 6. Report on the ISC general assembly 7. ICIAM 2019 congress 8. Abel Prize 2018 to Robert Langlands 9. 2018 Wolf prizes to Beilinson and Drinfeld 10. MSC 2020: Revision of Mathematics Subject Classification 11. Subscribing to IMU-Net ------------------------------------------------------------------------------------------------------- 1. EDITORIAL: ICM 2018 Dear Colleagues, After so many months of work and expectation, the year of the Congress has finally arrived! Preparations are well under way: the bulk of the scientific program has been defined, the proceedings are being finalized (the papers by plenary and invited speakers will be distributed to all the participants at the Congress), travel grants have been awarded to participants from the developing world, communications and posters are being selected as I write, and registration of participants is also taking place right now. Keep in mind that the deadline for early advance registration, with reduced registration fee, is April 27. Much of our effort over the last couple of years has been geared towards advertising the Congress, domestically and abroad, as we take the occasion as a historic opportunity to popularize mathematics in our society and, especially, amongst the Brazilian youth. I believe we are being very successful. The Biennium of Mathematics 2017-2018, formally proclaimed by the Brazilian national parliament, encompasses a wide range of outreach initiatives throughout the country, and raised the profile of mathematics in mainstream media to totally unprecedented levels. -
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 . -
Arxiv:2104.06941V2 [Math.AG] 26 May 2021
RIEMANN-HILBERT CORRESPONDENCE FOR ALEXANDER COMPLEXES LEI WU Abstract. We establish a Riemann-Hilbert correspondence for Alexander complexes (also known as Sabbah specialization complexes) by using relative holonomic D-modules in an equivariant way, which particularly gives a \global" approach to the correspondence for Deligne's nearby cycles. Using the corre- spondence and zero loci of Bernstein-Sato ideals, we obtain a formula for the relative support of the Alexander complexes. Contents 1. Introduction 1 2. Sheafification of sheaves of modules over commutative rings 8 3. Relative maximal and minimal extensions along F 17 4. Alexander complex of Sabbah 34 5. Comparison 35 References 40 1. Introduction Let f be a holomorphic function on a complex manifold X with D the divisor of f and let Ux be a small open neighborhood of x 2 D. Then we consider the fiber product diagram ∗ ∗ ∗ Uex = (Ux n D) ×C Ce Ce exp f ∗ Ux n D C ∗ ∗ ∗ arXiv:2104.06941v3 [math.AG] 1 Sep 2021 where exp: Ce ! C is the universal cover of the punctured complex plane C . ∗ The deck transformation induces a C[π1(C )]-module structure on the compactly i supported cohomology group Hc(Uex; C), which is called the i-th (local) Alexander module of f. Taking Ux sufficiently small, the Alexander modules contain the information of the cohomology groups of the Milnor fibers around x together with their monodromy action. As x varies along D, all the local Alexander modules ∗ give a constructible complex of sheaves of C[π1(C )]-modules, which recovers the Deligne nearby cycle of the constant sheaf along f (see [Bry86]). -
An Introduction to Algebraic D-Modules
AN INTRODUCTION TO ALGEBRAIC D-MODULES WYATT REEVES Abstract. This paper aims to give a friendly introduction to the theory of algebraic D-modules. Emphasis is placed on examples, computations, and intuition. The paper builds up the basic theory of D-modules, concluding with a proof of the Kashiwara equivalence of categories. We assume knowledge of basic homological algebra, algebraic geometry, and sheaf theory. Contents 1. Introduction 1 2. Differential Operators and D-modules 2 3. Derived Categories 3 4. Filtrations and Gradings 5 5. Pullback of D-modules 6 6. Pushforward of D-modules 9 Acknowledgments 15 References 15 1. Introduction Linearization is a wonderful thing. Let G be a Lie group acting on a smooth manifold X and let E ! X be a G-equivariant vector bundle over X. By generalizing the notion of a Lie derivative, we obtain an action of g on Γ(X; E): let s 2 Γ(X; E) and X 2 g. Define X · s 2 Γ(X; E) to be the section such that d (X · s)(p) = exp(Xt) · s (exp(−Xt) · p) : dt t=0 This action of g on Γ(X; E) should be thought of as the linearization of the action of G on E. The study of this linearized g action has historically been quite fruitful, leading for example to the resolution of the Kazhdan-Lusztig conjectures by Beilinson and Berstein [1]. A group action can be thought of as a map from a group G to the automorphisms of a space X. Analogously, a Lie algebra action can be thought of as a map of g into the derivations of X. -
Department of Mathematics 1
Department of Mathematics 1 Department of Mathematics Chair • Shmuel Weinberger Professors • Laszlo Babai, Computer Science and Mathematics • Guillaume Bal, Statistics and Mathematics • Alexander A. Beilinson • Danny Calegari • Francesco Calegari • Kevin Corlette • Marianna Csörnyei • Vladimir Drinfeld • Todd Dupont, Computer Science and Mathematics • Matthew Emerton • Alex Eskin • Benson Farb • Robert A. Fefferman • Victor Ginzburg • Denis Hirschfeldt • Kazuya Kato • Carlos E. Kenig • Gregory Lawler, Mathematics and Statistics • Maryanthe Malliaris • J. Peter May • Andre Neves • Bao Châu Ngô • Madhav Vithal Nori • Alexander Razborov, Mathematics and Computer Science • Luis Silvestre • Charles Smart • Panagiotis Souganidis • Sidney Webster • Shmuel Weinberger • Amie Wilkinson • Robert Zimmer Associate Professors • Simion Filip • Ewain Gwynne Assistant Professors • Sebastian Hurtado-Salazar • Dana Mendelson • Nikita Rozenblyum • Daniil Rudenko Instructors • Lucas Benigni • Guher Camliyurt • Stephen Cantrell • Elliot Cartee • Mark Cerenzia 2 Department of Mathematics • Andrea Dotto • Mikolaj Fraczyk • Pedro Gasper • Kornelia Hera • Trevor Hyde • Kasia Jankiewicz • Justin Lanier • Brian Lawrence • Zhilin Luo • Akhil Mathew • Henrik Matthieson • Cornelia Mihaila • Lucia Mocz • Benedict Morrissey • Davi Obata • Lue Pan • Wenyu Pan • Beniada Shabani • Danny Shi • Daniel Stern • Ao Sun • Xuan Wu • Zihui Zhao • Jinping Zhuge Senior Lecturers • John Boller • Lucas Culler • Jitka Stehnova • Sarah Ziesler Lecturer • Meghan Anderson Assistant Instructional -
The Hodge Theory of the Decomposition Theorem (After De
S´eminaire BOURBAKI Mars 2016 o 68`eme ann´ee, 2015-2016, n 1115 THE HODGE THEORY OF THE DECOMPOSITION THEOREM [after M. A. de Cataldo and L. Migliorini] by Geordie WILLIAMSON INTRODUCTION The Decomposition Theorem is a beautiful theorem about algebraic maps. In the words of MacPherson [Mac83], “it contains as special cases the deepest homological properties of algebraic maps that we know.” Since its proof in 1981 it has found spec- tacular applications in number theory, representation theory and combinatorics. Like its cousin the Hard Lefschetz Theorem, proofs appealing to the Decomposition The- orem are usually difficult to obtain via other means. This leads one to regard the Decomposition Theorem as a deep statement lying at the heart of diverse problems. The Decomposition Theorem was first proved by Beilinson, Bernstein, Deligne and Gabber [BBD]. Their proof proceeds by reduction to positive characteristic in order to use the Frobenius endomorphism and its weights, and ultimately rests on Deligne’s proof of the Weil conjectures. Some years later Saito obtained another proof of the Decomposition Theorem as a corollary of his theory of mixed Hodge modules [Sai87, Sai89]. Again the key is a notion of weight. More recently, de Cataldo and Migliorini discovered a simpler proof of the Decom- position Theorem [dCM02, dCM05]. The proof is an ingenious reduction to statements about the cohomology of smooth projective varieties, which they establish via Hodge theory. In their proof they uncover several remarkable geometric statements which go a arXiv:1603.09235v1 [math.AG] 30 Mar 2016 long way to explaining “why” the Decomposition Theorem holds, purely in the context of the topology of algebraic varieties. -
PARITY SHEAVES and the HECKE CATEGORY Introduction One of the First Theorems of Representation Theory Is Maschke's Theorem: An
PARITY SHEAVES AND THE HECKE CATEGORY GEORDIE WILLIAMSON Dedicated to Wolfgang Soergel, in admiration. \. we're still using your imagination. " (L. A. Murray) Introduction One of the first theorems of representation theory is Maschke's theorem: any representation of a finite group over a field of characteristic zero is semi-simple. This theorem is ubiquitous throughout mathematics. (We often use it without realising it; for example, when we write a function of one variable as the sum of an odd and an even function.) The next step is Weyl's theorem: any finite-dimensional representation of a compact Lie group is semi-simple1. It is likewise fundamental: for the circle group Weyl's theorem is closely tied to the theory of Fourier series. Beyond the theorems of Maschke and Weyl lies the realm of where semi-simplicity fails. Non semi-simple phenomena in representation theory were first encountered when studying the modular (i.e. characteristic p) representations of finite groups. This theory is the next step beyond the classical theory of the character table, and is important in understanding the deeper structure of finite groups. A second ex- ample (of fundamental importance throughout mathematics from number theory to mathematical physics) occurs when studying the infinite-dimensional representation theory of semi-simple Lie groups and their p-adic counterparts. Throughout the history of representation theory, geometric methods have played an important role. Over the last forty years, the theory of intersection cohomology and perverse sheaves has provided powerful new tools. To any complex reductive group is naturally associated several varieties (e.g. -
Derived Categories of Keum's Fake Projective Planes 1
DERIVED CATEGORIES OF KEUM'S FAKE PROJECTIVE PLANES SERGEY GALKIN, LUDMIL KATZARKOV, ANTON MELLIT, EVGENY SHINDER Abstract. We conjecture that derived categories of coherent sheaves on fake projective n-spaces have a semi-orthogonal decomposition into a collection of n + 1 exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum's surface). Then by passing to equivariant categories we construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing. Keywords: derived categories of coherent sheaves, exceptional collections, fake projective planes Mathematics Subject Classication 2010: 14F05, 14J29 1. Introduction and statement of results A fake projective plane is by definition a smooth projective surface with minimal cohomology (i.e. is the same cohomology as that of CP2) which is not isomorphic to CP2. Such surface is necessarily of general type. The first example of a fake projective plane was constructed by Mumford [43] using p-adic uniformization [20], [44]. Prasad{Yeung [47] and Cartwright{Steger [17] have recently finished classification of fake projective planes into 100 isomorphism classes. Due to an absense of a geometric construction for fake projective planes there are many open questions about them. Most notably the Bloch conjecture on zero-cycles [8] for fake projective planes is not yet established. In higher dimension we may call a smooth projective n-dimensional variety X of general type a fake projective space if it has minimal cohomology and in addition has the same \Hilbert poly- n ⊗l n ⊗l nomial" as : χ(X; ! ) = χ( ;! n ) for all l 2 . -
The Decomposition Theorem, Perverse Sheaves and the Topology of Algebraic Maps
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 4, October 2009, Pages 535–633 S 0273-0979(09)01260-9 Article electronically published on June 26, 2009 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 alge- braic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem and indicate some important applications and examples. Contents 1. Overview 536 1.1. The topology of complex projective manifolds: Lefschetz and Hodge theorems 538 1.2. Families of smooth projective varieties 538 1.3. Singular algebraic varieties 540 1.4. Decomposition and hard Lefschetz in intersection cohomology 540 1.5. Crash course on sheaves and derived categories 541 1.6. Decomposition, semisimplicity and relative hard Lefschetz theorems 545 1.7. Invariant cycle theorems 547 1.8. A few examples 548 1.9. The decomposition theorem and mixed Hodge structures 549 1.10. Historical and other remarks 550 2. Perverse sheaves 551 2.1. Intersection cohomology 552 2.2. Examples of intersection cohomology 553 2.3. Definition and first properties of perverse sheaves 555 2.4. The perverse filtration 558 2.5. Perverse cohomology 559 2.6. t-exactness and the Lefschetz hyperplane theorem 560 2.7. Intermediate extensions 561 3.