Geometry of Moduli Spaces and Representation Theory 9781470435745

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IAS/PARK CITY MATHEMATICS SERIES Volume 24

Geometry of Moduli Spaces and Representation Theory Roman Bezrukavnikov Alexander Braverman Zhiwei Yun Editors

American Mathematical Society Institute for Advanced Study Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

10.1090/pcms/024

Geometry of Moduli Spaces and Representation Theory

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IAS/PARK CITY

MATHEMATICS SERIES Volume 24

Geometry of Moduli Spaces and Representation Theory Roman Bezrukavnikov Alexander Braverman Zhiwei Yun Editors

American Mathematical Society Institute for Advanced Study

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Rafe Mazzeo, Series Editor Roman Bezrukavnikov, Alexander Braverman, and Zhiwei Yun, Volume Editors. The IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program. 2010 Mathematics Subject Classification. Primary 14N35, 14M17, 14D24, 22E57, 22E67.

Library of Congress Cataloging-in-Publication Data Names: Bezrukavnikov, Roman, 1973– editor. | Braverman, Alexander, 1974– editor. | Yun, Zhiwei, 1982– editor. Title: Geometry of moduli spaces and representation theory / Roman Bezrukavnikov, Alexander Braverman, Zhiwei Yun, editors. Description: Providence : American Mathematical Society ; [Princeton, New Jersey] : Institute for Advanced Study, 2017. | Series: IAS/Park City mathematics series ; volume 24 | Includes bibliographical references. Identifiers: LCCN 2017018956 | ISBN 9781470435745 (alk. paper) Subjects: LCSH: Mathematics–Problems, exercises, etc. | Geometry, Algebraic–Problems, exercises, etc. | Moduli theory–Problems, exercises, etc. | Algebraic varieties–Problems, exercises, etc. | AMS: Algebraic geometry – Projective and enumerative geometry – Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants. msc | Algebraic geometry – Special varieties – Homogeneous spaces and generalizations. msc | Algebraic geometry – Families, fibrations – Geometric Langlands program: algebro-geometric aspects. msc | Topological groups, Lie groups – Lie groups – Geometric Langlands program: representationtheoretic aspects. msc | Topological groups, Lie groups – Lie groups – Loop groups and related constructions, group-theoretic treatment. msc Classification: LCC QA565 .G4627 2017 | DDC 516.3/5–dc23 LC record available at https:// lccn.loc.gov/2017018956

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Contents Preface

ix

Introduction

xi

Perverse sheaves and the topology of algebraic varieties Mark Andrea de Cataldo

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An introduction to affine Grassmannians and the geometric Satake equivalence Xinwen Zhu 59 Lectures on Springer theories and orbital integrals Zhiwei Yun

155

Perverse sheaves and fundamental lemmas Ngˆ o Bảo Chˆ au

217

Lectures on K-theoretic computations in enumerative geometry Andrei Okounkov

251

Lectures on perverse sheaves on instanton moduli spaces Hiraku Nakajima

381

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IAS/Park City Mathematics Series Volume 24, Pages -7—8 http://dx.doi.org/10.1090/pcms/024/00821

Preface The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the Regional Geometry Institute initiative of the National Science Foundation. In mid-1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and education in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, graduate students, undergraduate students, high school students, undergraduate faculty, K-12 teachers, and international teachers and education researchers. The Teacher Leadership Program also includes weekend workshops and other activities during the academic year. One of PCMI’s main goals is to make all of the participants aware of the full range of activities that occur in research, mathematics training and mathematics education: the intention is to involve professional mathematicians in education and to bring current concepts in mathematics to the attention of educators. To that end, late afternoons during the summer institute are devoted to seminars and discussions of common interest to all participants, meant to encourage interaction among the various groups. Many deal with current issues in education: others treat mathematical topics at a level which encourages broad participation. Each year the Research Program and Graduate Summer School focuses on a different mathematical area, chosen to represent some major thread of current mathematical interest. Activities in the Undergraduate Summer School and Undergraduate Faculty Program are also linked to this topic, the better to encourage interaction between participants at all levels. Lecture notes from the Graduate Summer School are published each year in this series. The prior volumes are: • Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Differential Equations in Differential Geometry (1992) • Volume 3: Complex Algebraic Geometry (1993) • Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) ©2017 American Mathematical Society

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Preface

• Volume 10: • Volume 11: try (2001) • Volume 12: • Volume 13: • Volume 14: • Volume 15: • Volume 16: • Volume 17: • Volume 18: • Volume 19: • Volume 20: • Volume 21: • Volume 22: • Volume 23:

Computational Complexity Theory (2000) Quantum Field Theory, Supersymmetry, and Enumerative GeomeAutomorphic Forms and their Applications (2002) Geometric Combinatorics (2004) Mathematical Biology (2005) Low Dimensional Topology (2006) Statistical Mechanics (2007) Analytical and Algebraic Geometry (2008) Arithmetic of L-functions (2009) Mathematics in Image Processing (2010) Moduli Spaces of Riemann Surfaces (2011) Geometric Group Theory (2012) Geometric Analysis (2013) Mathematics and Materials (2014)

The American Mathematical Society publishes material from the Undergraduate Summer School in their Student Mathematical Library and from the Teacher Leadership Program in the series IAS/PCMI—The Teacher Program. After more than 25 years, PCMI retains its intellectual vitality and continues to draw a remarkable group of participants each year from across the entire spectrum of mathematics, from Fields Medalists to elementary school teachers. Rafe Mazzeo PCMI Director March 2017

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IAS/Park City Mathematics Series Volume 24, Pages -9—10 http://dx.doi.org/10.1090/pcms/024/00820

Introduction Roman Bezrukavnikov, Alexander Braverman, and Zhiwei Yun The 2015 Park City Mathematics Institute program on “Geometry of moduli spaces and representation theory” was devoted to a combination of interrelated topics in algebraic geometry, topology of algebraic varieties and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of the those subjects. An early profound achievement was the formulation, in the late 70’s, of Kazhdan and Lusztig’s famous conjecture on characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson–Bernstein and Brylinski–Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. The topological invariants in question have to do with intersection cohomology of Schubert varieties, while the key algebro-geometric result used in the proof is a generalization of Weil’s conjecture by Beilinson, Bernstein and Deligne involving perverse sheaves. The geometric spaces appearing in the Kazhdan–Lusztig conjectures are closed subvarieties in the flag variety, a homogeneous space which is a basic ingredient in the theory of algebraic groups. A later major direction in geometric representation theory, shaped by contributions of Lusztig, Nakajima and others, develops a similar relation between representation theory and moduli spaces of linear algebra data (quiver varieties). More intricate geometric objects have entered the subject with the emergence of the geometric Langlands program. This direction, pioneered by Beilinson and Drinfeld in the 90’s, is partly inspired by Langlands’ conjectural nonabelian reciprocity laws from number theory. In the last decade, Kapustin and Witten have discovered its close connection to S-duality in quantum field theory. While employing some of the techniques of Kazhdan-Lusztig theory, geometric Langlands duality deals with more sophisticated geometric spaces, such as the moduli space (or stack) of principal bundles on a complete algebraic curve and its local counterpart, the affine Grassmannian, also known as the loop Grassmannian. A large part of the PCMI program was devoted to introducing this circle of ideas. Another focus of the program was on some aspects of enumerative algebraic geometry. Recent progress in that area has been increasingly bringing to light ©2017 American Mathematical Society

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the role of Lie theoretic structures in problems such as calculation of (equivariant) quantum cohomology, K-theory etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The goal of the program was to provide an introduction to these areas of active research and promote interaction between the two related directions. Our hope is that this will help to write a new chapter in the glorious history of the interaction between representation theory and algebraic geometry. Just as D-modules, perverse sheaves and the generalizations of Weil’s conjecture have become standard tools in studying many algebraic questions in representation theory, we hope that keys to resolving other outstanding questions may lie in the recent techniques of enumerative algebraic geometry The program included minicourses by Alexander Braverman, Mark de Cataldo, Victor Ginzburg, Davesh Maulik, Hiraku Nakajima, Xinwen Zhu, Zhiwei Yun, and Clay Scholars Ngô Bảo Châu and Andrei Okounkov. This volume contains contributions by Mark de Cataldo, Hiraku Nakajima, Ngô Bảo Châu, Andrei Okounkov, Xinwen Zhu and Zhiwei Yun.

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10.1090/pcms/024/01 IAS/Park City Mathematics Series Volume 24, Pages 1–58 http://dx.doi.org/10.1090/pcms/024/00814

Perverse sheaves and the topology of algebraic varieties Mark Andrea de Cataldo Dedicato a Mikki

Contents Introduction Lecture 1: The decomposition theorem 1.1 Deligne’s theorem in cohomology 1.2 The global invariant cycle theorem 1.3 Cohomological decomposition theorem 1.4 The local invariant cycle theorem 1.5 Deligne’s theorem 1.6 The decomposition theorem 1.7 Exercises for Lecture 1 Lecture 2: The category of perverse sheaves P(Y) 2.1 Three "Whys", and a brief history of perverse sheaves 2.2 The constructible derived category D(Y) 2.3 Definition of perverse sheaves 2.4 Artin vanishing and Lefschetz hyperplane theorems 2.5 The perverse t-structure 2.6 Intersection complexes 2.7 Exercises for Lecture 2 Lecture 3: Semi-small maps 3.1 Semi-small maps 3.2 The decomposition theorem for semi-small maps 3.3 Hilbert schemes of points on surfaces and Heisenberg algebras 3.4 The endomorphism algebra End(f∗ QX ) 3.5 Geometric realization of the representations of the Weyl group 3.6 Exercises for Lecture 3 Lecture 4: Symmetries: VD, RHL, IC splits off 4.1 Verdier duality and the decomposition theorem 4.2 Verdier duality and the decomposition theorem with large fibers 4.3 The relative hard Lefschetz theorem

2 3 3 4 5 6 7 8 10 17 17 19 21 22 24 25 26 28 29 31 31 33 34 35 37 37 38 39

Partially supported by N.S.F. grant DMS-1301761 and by a grant from the Simons Foundation (#296737 to Mark de Cataldo). ©2017 American Mathematical Society

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4.4 Application of RHL: Stanley’s theorem 4.5 Intersection cohomology of the target as a direct summand 4.6 Pure Hodge structure on intersection cohomology groups 4.7 Exercises for Lecture 4 Lecture 5: The perverse filtration 5.1 The perverse spectral sequence and the perverse filtration 5.2 Geometric description of the perverse filtration 5.3 Hodge-theoretic consequences 5.4 Character variety and Higgs moduli: P = W 5.5 Let us conclude with a motivic question 5.6 Exercises for Lecture 5

40 41 43 44 46 47 48 50 50 53 54

Introduction Goal of the lectures. The goal of these lectures is to introduce the novice to the use of perverse sheaves in complex algebraic geometry and to what is perhaps the deepest known fact relating the homological invariants of the source and target of a proper map of complex algebraic varieties, namely the decomposition theorem. Notation. A variety is a complex algebraic variety, which we do not assume to be irreducible, nor reduced. We work with cohomology with Q-coefficients as Z-coefficients do not fit well in our story. As we rarely focus on a single cohomological degree, for the most part we consider the total, graded cohomology groups, which we denote by H∗ (X, Q). Bibliographical references. The main reference is the survey [19] and the extensive bibliography contained in it, most of which is not reproduced here. This allowed me to try to minimize the continuous distractions related to the peeling apart of the various versions of the results and of the attributions. The reader may also consult the discussions in [18] that did not make it into the very different final version [19]. Style of the lectures and of the lecture notes. I hope to deliver my lectures in a rather informal style. I plan to introduce some main ideas, followed by what I believe to be a striking application, often with an idea of proof. The lecture notes are not intended to replace in any way the existing literature on the subject, they are a mere amplification of what I can possibly touch upon during the five onehour lectures. As it is usual when meeting a new concept, the theorems and the applications are very important, but I also believe that working with examples, no matter how lowly they may seem, can be truly illuminating and useful in building one’s own local and global picture. Because of the time factor, I cannot possibly fit many of these examples in the flow of the lectures. This is why there are plenty of exercises, which are not just about examples, but at times deal headon with actual important theorems. I could have laid-out several more exercises

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(you can look at my lecture notes [22], or at my little book [9] for more exercises), but I tried to choose ones that would complement well the lectures; too much of anything is not a good thing anyway. What is missing from these lectures? A lot! Two related topics come to mind: vanishing/nearby cycles and constructions of perverse sheaves; see the survey [19] for a quick introduction to both. To compound this infamy, there is no discussion of the equivariant picture [3]. An afterthought. The 2015 PCMI is now over. Even though I have been away from Mikki, Caterina, Amelie (Amie!) and Dylan for three weeks, my PCMI experience has been wonderful. If you love math, then you should consider participating in future PCMIs. Now, let us get to Lecture 1.

. Lecture 1: The decomposition theorem Summary of Lecture 1. Deligne’s theorem on the degeneration of the Leray spectral sequence for smooth projective maps; this is the 1968 prototype of the 1982 decomposition theorem. Application, via the use of the theory of mixed Hodge structures, to the global invariant cycle theorem, a remarkable topological property enjoyed by families of projective manifolds and compactifications of their total spaces. The main theorem of these lectures, the decomposition theorem, stated in cohomology. Application to a proof of the local invariant cycle theorem, another remarkable topological property concerning degenerations of families of projective manifolds. Deligne’s theorem, including semi-simplicity of the direct image sheaves, in the derived category. The decomposition theorem: the direct image complex splits in the derived category into a direct sum of shifted and twisted intersection complexes supported on the target of a proper map. 1.1. Deligne’s theorem in cohomology Let us start with a warm-up: the Künneth formula and a question. Let Y, F be varieties. Then  H∗−q (Y, Q) ⊗ Hq (F, Q). (1.1.1) H∗ (Y × F, Q) = q0

H∗ (Y

Note that the restriction map × F, Q) → H∗ (F, Q) is surjective. This surjectivity fails in the context of (differentiable) fiber bundles: take the Hopf fibration b : S3 → S2 (cf. Exercise 1.7.2), for example. It is a remarkable fact that, in the context of algebraic geometry, one has indeed this surjectivity property, and more. Let us start discussing this phenomenon by asking the following Question 1.1.2. Let f : X → Y be a family of projective manifolds. What can we say about the restriction maps H∗ (X, Q) → H∗ (f−1 (y), Q)? Let X be a projective manifold completing X (i.e. X is open and Zariski-dense in X). What can we say about the restriction maps H∗ (X, Q) → H∗ (f−1 (y), Q)?

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Answer: The answers are given, respectively, by (1.2.1) and by the global invariant cycle Theorem 1.2.2. Both rely on Deligne’s Theorem, which we review next. The decomposition theorem has an important precursor in Deligne’s theorem, which can be viewed as the decomposition theorem in the absence of singularities of the domain, of the target and of the map. We start by stating the cohomological version of his theorem. Theorem 1.1.3. (Blanchard-Deligne 1968 theorem in cohomology [24]) For any smooth projective map1 f : X → Y of algebraic manifolds, there is an isomorphism  ∼ H∗−q (Y, Rq f∗ QX ), (1.1.4) H∗ (X, Q) = q0 ∗ Q X denotes the q-th direct image sheaf of the sheaf Q X via the morphism f; see §1.2. More precisely, the Leray spectral sequence (see §1.7) of the map f is E2 -degenerate.

where Rq f

Proof. Exercise 1.7.3 guides you through Deligne’s classical trick (the DeligneLefschetz criterion) of using the hard Lefschetz theorem on the fibers to force the triviality of the differentials of the Leray spectral sequence.  Compare (1.1.1) and (1.1.4): both present cohomological shifts; both express the cohomology of the l.h.s. via cohomology groups on Y; in the former case, we have cohomology with constant coefficients; in the latter, and this is an important distinction, we have cohomology with locally constant coefficients. Deligne’s theorem is central in the study of the topology of algebraic varieties. Let us discuss one striking application of this result: the global invariant cycle theorem. 1.2. The global invariant cycle theorem Let f : X → Y be a smooth and projective map of algebraic manifolds, let j : X → X be an open immersion into a projective manifold and let y ∈ Y. What are the images of H∗ (X, Q) and H∗ (X, Q) via the restriction maps into H∗ (f−1 (y), Q)? The answer is the global invariant cycle Theorem 1.2.2 below. The direct image sheaf Rq := Rq f∗ QX on Y is the sheaf associated with the pre-sheaf U → Hq (f−1 (U), Q). In view of Ehresmann’s lemma, the proper2 submersion f is a C∞ fiber bundle. The sheaf Rq is then locally constant with stalk q −1 Rq y = H (f (y), Q).

The fundamental group π1 (Y, y) acts via linear transformations on Rq y : pick a loop γ(t) at y and use a trivialization of the bundle along the loop to move vectors in q q q Rq y along Rγ(t) , back to Ry (monodromy action for the locally constant sheaf R ). submersion; projective: factors as X → Y × P → Y (closed embedding, projection). := the pre-image of compact is compact; it is the “relative” version of compactness.

1 Smooth: 2 Proper

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q π Global sections of Rq identify with the monodromy invariants (Rq y ) 1 ⊆ Ry . Note, further, that this subspace is defined topologically. The cohomology group q −1  Rq y = H (f (y), Q) has it own Hodge (p, p )-decomposition (pure Hodge structure of weight q), an algebro-geometric structure. q π How is (Rq y ) 1 ⊆ Ry placed with respect to the Hodge structure? The E2 -degeneration Theorem 1.1.3 yields the following immediate, yet, remarkable, consequence:

(1.2.1)

surj

π1 q −1 ⊆ Rq Hq (X, Q) −→ (Rq y) y = H (f (y), Q),

i.e. the restriction map in cohomology, which automatically factors through the invariants, maps surjectively onto them. The theory of mixed Hodge structures now tells us that the monodromy invariq π ant subspace (Rq y ) 1 ⊆ Ry (a topological gadget) is in fact a Hodge substructure, i.e. it inherits the Hodge (p, p  )-decomposition (the algebro-geometric gadget). The same mixed theory implies the highly non-trivial fact (Exercise 1.7.14) that the images of the restriction maps from H∗ (X, Q) and H∗ (X, Q) into H∗ (f−1 (y), Q) coincide. We have reached the following conclusion, proved by Deligne in 1972. Theorem 1.2.2. (Global invariant cycle theorem [26]) Let f : X → Y be a smooth and projective map of algebraic manifolds, let j : X → X be an open immersion into a projective manifold and let y ∈ Y. Then the images of H∗ (X, Q) and H∗ (X, Q) into H∗ (f−1 (y), Q) coincide with the subspace of monodromy invariants. In particular, this latter is a Hodge substructure of the pure Hodge structure Hq (f−1 (y), Q). This theorem provides a far-reaching answer to Question 1.1.2. Note that the Hopf examples in Exercise 1.7.2 show that such a nice general answer is not possible outside of the realm of complex algebraic geometry: there are two obstacles, i.e. the non E2 -degeneration, and the absence of the special kind of global constraints imposed by mixed Hodge structures. 1.3. Cohomological decomposition theorem The decomposition theorem is a generalization of Deligne’s Theorem 1.1.3 for smooth proper maps to the case of arbitrary proper maps of algebraic varieties: compare (1.1.4) and (1.3.2). It was first proved by Beilinson-Bernstein-DeligneGabber in their monograph [2, Théorème 6.2.5] on perverse sheaves. A possible initial psychological drawback, when compared with Deligne’s theorem, is that even if one insists in dealing with maps of projective manifolds, the statement is not about cohomology with locally constant coefficients, but requires the Goresky-MacPherson intersection cohomology groups with twisted coefficients on various subvarieties of the target of the map. However, this is precisely why this theorem is so striking! To get to the point, for now we simply say that we have the intersection cohomology groups IH∗ (S, Q) of an irreducible variety S; they agree with the ordinary

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Perverse sheaves and the topology of algebraic varieties

cohomology groups when S is nonsingular. The theory is very flexible as it allows for twisted coefficients: given a locally constant sheaf L on a dense open subvariety So ⊆ Sreg ⊆ S, we get the intersection cohomology groups IH∗ (S, L) of S. We may call such pairs (S, L), enriched varieties (see [39, p.222]; this explains the notation EV below. Theorem 1.3.1. (Cohomological decomposition theorem) Let f : X → Y be a proper map of complex algebraic varieties. For every q  0, there is a finite collection EVq of pairs (S, L) with S ⊆ Y pairwise distinct closed subvarieties of Y, and an isomorphism   ∼ IH∗−q (S, L). (1.3.2) IH∗ (X, Q) = q0 (S,L)∈EVq

Note that the same S could appear for distinct q’s, hence the notation EVq . Deligne’s Theorem 1.1.3 in cohomology is a special case. In particular, we can deduce an appropriate version of the global invariant cycle theorem [2, 6.2.8]. Let us instead focus on its local counterpart. 1.4. The local invariant cycle theorem The decomposition theorem (1.3.2) has a local flavor over the target Y, in both the Zariski and in the classical topology: if we replace Y by an open set U ⊆ Y, X by f−1 (U), and S by S ∩ U, then (1.3.2) remains valid. Let us focus on the classical topology. Let X be nonsingular; this is for the sake of our discussion, for then IH∗ (X, Q) = H∗ (X, Q). Let y ∈ Y be a point and let us pick a small Euclidean “ball” By ⊆ Y centered at y, so that (1.3.2) reads:  IH∗−q (S ∩ By , L), H∗ (f−1 (y), Q) = H∗ (f−1 (By ), Q) = q0,(S,L)∈EVq

where the second equality stems directly from (1.3.2), and the first one can be seen as follows: the constructibility of the direct image complex Rf∗ QX ensures that the second term can be identified with the stalk (R∗ f∗ QX )y , and, in turn, the proper base change theorem ensures that this latter is the first term; see Fact 2.2.1. Let f be surjective. Let fo : Xo → Y o the restriction of the map f over the open subvariety of Y of regular values for f. Let yo ∈ By be a regular value for f. By looking at Deligne theorem for the map fo it seems reasonable to expect that for every q one of the summands in (1.3.2) should be IH∗−q (Y, Lq ), where Lq is the locally constant sheaf Rq fo ∗ Q. This is indeed the case. If follows that for every q  0, IH0 (By , Lq |Y o ∩By ) is a direct summand of q H (f−1 (y), Q), let us even say that the latter surjects onto the former. Note that we did not assume that y ∈ Y o . The intersection cohomology group IH0 (Y, Lq ) is the space of monodromy invariants for the representation π1 (Y o ∩ By , yo ) → GL(Hq (f−1 (yo ), Q). Abbreviate the fundamental group notation to π1,loc . We have reached a very important conclusion:

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Theorem 1.4.1. (Local invariant cycle theorem, [7] and [2, (6.2.9)] Let f : X → Y be a proper surjective map of algebraic varieties with X nonsingular. Let y ∈ Y be any point, let By be a small Euclidean ball on Y at y, let yo ∈ By be a regular value of f. Then H∗ (f−1 (y), Q) = H∗ (f−1 (By ), Q) surjects onto the local monodromy invariants H∗ (f−1 (yo ), Q)π1,loc . 1.5. Deligne’s theorem In fact, Deligne proved something stronger than his cohomological theorem (1.1.4), he proved a decomposition theorem for the derived direct image complex under a smooth proper map. Pre-warm-up: cohomological shifts. Given a Z-graded object K = ⊕i∈Z Ki , like the total cohomology of a variety, or a complex (of sheaves, for example) on it, or the total cohomology of such a complex, etc., and given an integer a ∈ Z, we can shift by the amount a and get a new graded object (with Ki+a in degree i)  Ki+a . (1.5.1) K[a] := i∈Z

If a > 0, then the effect of this operation is to “shift K back by a units.” Again, if K has non zero entries contained in an interval [m, n], then K[a] has non zero entries contained in [m − a, n − a]. We have the following basic relation, e.g. for complexes of sheaves Hi (K[a]) = Hi+a (K). A sheaf F can be viewed as a complex placed in cohomological degree zero; we can then take the F[a]’s. We can take a collection of Fq ’s and form ⊕q Fq [−q], which is a complex with trivial differentials. Then H∗ (Y, ⊕q Fq [−q]) = ⊕q H∗−q (Y, Fq ). Warm-up: Künneth for the derived direct image. Let f : X := Y × F → Y be the projection. Then there is a canonical isomorphism (1.5.2)

Rf∗ QX = ⊕q0 Hq (F)[−q]

where Hq (F) is the constant sheaf on Y with stalk Hq (F, Q). The isomorphism takes place in the derived category of the category of sheaves of rational vector spaces on Y; this is where we find the direct image complex Rf∗ QX , whose cohomology is the cohomology of X: H∗ (Y, Rf∗ QX ) = H∗ (X, Q). Exercise 1.7.23 asks you to prove (1.5.2). Now to Deligne’s 1968 theorem. Theorem 1.5.3. (Deligne’s 1968 theorem [24]; semi-simplicity in 1972 [26, §4.2] ) Let f : X → Y be a smooth proper map of algebraic varieties. “The derived direct image complex has trivial differentials”, more precisely, there is an isomorphism in the derived category  ∼ Rq f∗ Q [−q]. (1.5.4) Rf∗ QX = q0

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Perverse sheaves and the topology of algebraic varieties

Moreover, the locally constant direct image sheaves Rq f∗ QX are semi-simple. Equation (1.5.4), which is proved by means of an E2 -degeneration argument3 along the lines of the one in Exercise 1.7.3, is the “derived” version of (1.1.4), which follows by taking cohomology on both sides of (1.5.4). In addition to [24], you may want to consult the first two pages of [30]. The semi-simplicity result is one of the many amazing applications of the theory of weights (Hodge-theoretic, or Frobenius). Terminology and facts about semi-simple locally constant sheaves. To give a locally constant sheaf on Y is the same as giving a representation of the fundamental group of Y (Exercise 1.7.10). By borrowing from the language of representations, we have the notions of simple (no non-trivial locally constant subsheaf; a.k.a. irreducible) and semi-simple (direct sum of simples; a.k.a. completely reducible), indecomposable (no non-trivial direct sum decomposition) locally constant sheaves. Once one has semi-simplicity, one can decompose further. For a semi-simple locally constant sheaf L, we have the canonical isotypic direct sum decomposition (1.5.5)

L = ⊕χ L χ ,

where each summand is the span of all mutually isomorphic simple subobjects, and the direct sum ranges over the set of isomorphism classes of irreducible representations of the fundamental group. In particular, in (1.5.4), we have Rq f∗ QX = Rq = ⊕χ Rq χ. What is semi-simplicity good for? Here is the beginning of an answer: look at Exercise 4.7.4, where it is put to good use to give Deligne’s proof in [29] of the Hard Lefschetz theorem. In the context of the decomposition theorem, the semi-simplicity of the perverse direct images is an essential ingredient in the proof of the relative hard Lefschetz theorem; see [2] and [16, especially, §5.1 and §6.4]. Keep in mind that the local systems arising naturally in algebraic geometry are typically not semisimple (cf. G. Williamson’s Math Overflow post on "non semi-simple monodromy in an algebraic family"). 1.6. The decomposition theorem As we have seen, Deligne’s theorem in cohomology has a counterpart in the derived category. The cohomological decomposition Theorem 1.3.1 also has a stronger counterpart in the derived category, i.e Theorem 1.6.2. In these lectures, we adopt a version of the decomposition theorem that is more general (and simpler to state!) than the one in [2, 6.2.5] (coefficients of geometric origin) and of [48] (coefficients in polarizable variations of pure Hodge structures). The version we adopt is due essentially to T. Mochizuki [41, 42] (with important contributions of C. Sabbah [47]) and it involves semi-simple coefficients. Mochizuki’s [41, 42] deals with projective maps of quasi-projective varieties and 3 People

refer to it as the Deligne-Lefschetz criterion.

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with C-coefficients; one needs a little bit of tinkering to reach the same conclusions for proper maps of complex varieties with Q-coefficients (to my knowledge, this is not in the literature). Warning: IC vs. IC. We are about to meet the main protagonists of our lectures, the intersection complexes ICS (L) with twisted coefficients; in fact, the actual protagonists are the shifted (see (1.5.1) for the notion of shift): (1.6.1)

ICS (L) := ICS (L)[dim S],

which are perverse sheaves on S and on any variety Y for which S ⊆ Y is closed. Note that ICS (L) only has non-trivial cohomology sheaves in the interval [0, dim S − 1], the analogous interval for ICS (L) is [− dim S, −1]. Both IC and IC are called intersection complexes. Instead of discussing the pros and cons of either notation, let us move on. Brief on intersection complexes. The intersection cohomology groups of an enriched variety (S, L) are in fact the cohomology groups of S with coefficients in a very special complex of sheaves called the intersection complex of S with coefficients in L and denoted by ICS (L): we have IH∗ (S, L) = H∗ (S, ICS (L)). If S is nonsingular, and L is constant of rank one, then ICS = ICS (Q) = QS . The decomposition theorem in cohomology (1.3.2) is the shadow in cohomology of a decomposition of the direct image complex Rf∗ ICX in the derived category of sheaves of rational vector spaces on Y. In fact, the decomposition theorem holds in the greater generality of semi-simple coefficients. Theorem 1.6.2. (Decomposition theorem) Let f : X → Y be a proper map of complex algebraic varieties. Let ICX (M) be the intersection complex of X with semi-simple twisted coefficients M. For every q  0, there is a finite collection EVq of pairs (S, L) with S pairwise distinct4 and L semi-simple, and an isomorphism  ∼ (1.6.3) Rf∗ ICX (M) = q0 (S,L)∈EVq ICS (L)[−q]. In particular, by taking cohomology: (1.6.4)

∼ IH∗ (X, M) =



IH∗−q (S, L).

q,EVq

We have the isotypic decompositions (1.5.5), which can be plugged into (1.6.4). Remark 1.6.5. The fact that there may be summands associated with S = Y should not come as a surprise. It is a natural fact arising from to the singularities (deviation from being smooth) of the map f. One does not need the decomposition theorem to get convinced: the reader can work out the case of the blowing up of the affine plane at the origin; see also Exercise 1.7.20. In general, it is difficult to predict which S will appear in the decomposition theorem; see parts 5 and 7 of Exercise 1.7.21. 4 The

same S could appear for distinct q’s.

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1.7. Exercises for Lecture 1 Exercise 1.7.1. (Ehresmann’s lemma and local constancy of higher direct images for proper submersions) Let f : X → Y be a map of varieties and recall that the q-th direct image sheaf Rq := Rq f∗ QX is defined to be the sheafification of the presheaf Y ⊇ U → Hq (f−1 (U), Q). If f admits the structure of a C∞ fiber bundle, then the sheaves Rq are locally constant, with stalks the cohomology of the fibers. Give examples of maps f : X → Y, where the stalks (Rq f∗ QX )y of the direct image of the constant sheaf differ from the cohomology groups H∗ (f−1 (y), Q) of the corresponding fibers (hint: the maps cannot be proper). If f is a proper smooth map of complex algebraic varieties, then it admits a structure of C∞ fiber bundle (Ehresmann’s lemma). Deduce that nonsingular hypersurfaces of fixed degree in complex projective space are all diffeomorphic to each other. Is the same true in real projective space? Why? Quick review of the Leray spectral sequence (see Grothendieck’s gem [34], a.k.a. “Tohoku”). The Leray spectral sequence for a map f : X → Y (and for = Hp (Y, Rq f∗ QX ) ⇒ Hp+q (X, Q). There the sheaf QX ) is a gadget denoted Epq 2 pq p+r,q−r+1 , d2r = 0, with r  2 and for which are the differentials dr : Er → Er Er+1 = H∗ (Er , dr ). E2 -degeneration means that dr = 0 for every r  2, so ∼ ⊕q0 H∗−q (Y, Rq f∗ QX ). that one has a cohomological decomposition H∗ (X, Q) = Note that with Z coefficients, E2 -degeneration does not imply the existence of an analogous splitting. ∼ S2 be the usual map Exercise 1.7.2. (Maps of Hopf-type) Let a : C2 \ o → P1 = (x, y) → (x : y) from the affine plane with the origin o deleted onto the projective line. It induces maps, b : S3 → S2 and c : HS := (C2 \ o)/Z → P1 (where 1 ∈ Z acts as multiplication by two). These three maps are fiber bundles. Show that there cannot be a cohomological decomposition as in (1.1.4). Deduce that their Leray spectral sequences are not E2 -degenerate. Observe that the conclusion of the global invariant cycle theorem concerning the surjectivity onto the monodromy invariants fails in all three cases. Exercise 1.7.3. (Proof of the cohomological decomposition (1.1.4) via hard Lefschetz) Let us recall the hard Lefschetz theorem: let X be a projective manifold of dimension d, and let η ∈ H2 (X, Q) be the first Chern class of an ample line bundle on X; then the iterated cup product maps ηd−q : Hq (X, Q) → H2d−q (X, Q) are isomorphisms for every q  0. Deduce the primitive Lefschetz decomposid−q+1 : Hq → H2d−q+2 }; then, for tion: for every q  d, set Hq prim := Ker {η q−2j q every 0  q  d, we have H = ⊕j0 Hprim , and, for d  q  2d, we have ). Let f : X → Y be as in (1.1.4), i.e. smooth and Hq = ηq−d ∪ (⊕j0 Hq−2d−2j prim projective and let d := dim X − dim Y. Apply the hard Lefschetz theorem to the fibers of the smooth map f and deduce the analogue of the primitive Lefschetz decomposition for the direct image sheaves Rq := Rf∗ QX . Argue that in order to deduce (1.1.4) it is enough to show the differentials dr of the spectral sequence

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vanish on Hp (Y, Rq prim ) for every q  d. Use the following commutative diagram, with some entries left blank on purpose for you to fill-in, to deduce that indeed we have the desired vanishing: H? (Y, Rq prim )

d?

η?

 H? (Y, ?)

/ H? (Y, ?) η?

d?

 / H? (Y, ?).

(Hint: the right power of η kills a primitive class in degree q, but is injective in degree q − 1.) Remark: the refined decomposition (1.5.4) is proved in a similar way by replacing the spectral sequence above with the analogous one for Hom(Rq [−q], Rf∗ QX ): first you prove it is E2 -degenerate; then you lift the identity Rq → Rq to a map in Hom(Rq [−q], Rf∗ QX ) inducing the identity on Rq ; see [30]. Heuristics for E2 -degeneration and for semi-simplicity of the Rq f∗ QX via weights. (What follows should be considered as a very informal fireside chat.) It seems that Deligne guessed at E2 -degeneration by looking at the same situation over the algebraic closure of a finite field by considerations (“the yoga of weights” [25, 28]) of the size (weight) of the eigenvalues of the action of Frobenius on the entries Epq r : they should have weight something analogous to exp (p + q) (we are using the exponential function as an analogy only, one needs to say more, but we shan’t) so the Frobenius-compatible differentials must be zero. There are similar heuristic considerations for Deligne’s theorem to the effect that the Rq are semisimple: if 0 → M → Rq → N → 0 is a short exact sequence, then Frobenius acts on Ext1 (N, M) with weight exp (1); take M ⊆ Rq to be the maximal semi-simple subobject; then the corresponding extension is invariant under Frobenius and has weight zero (1 = exp (0)); it follows that the extension splits and the biggest ∼ M ⊕ N; if the resulting quotient N were nonsemi-simple in Rq splits off: Rq = trivial, then it would contain a non-trivial simple that then, by the splitting, would enlarge the biggest semi-simple M in Rq ; contradiction. This kind of heuristic is now firmly based in deep theorems by Deligne and others [2, 29] for varieties over finite fields and their algebraic closure, and by M. Saito [48] in the context of mixed Hodge modules over complex algebraic varieties. Exercise 1.7.4. (Rank one locally constant sheaves) Take [0, 1] × Q and identify the two ends, {0} × Q and {1} × Q, by multiplication by −1 ∈ Q∗ . Interpret this as a rank one locally constant sheaf on S1 that is not constant. Do the same, but multiply by 2. Do the same, but first replace Q with Q and multiply by a root of unity. Show that the tensor product operation (L, M) → L ⊗ M induces the structure of an abelian group on the set of isomorphisms classes of rank one locally constant sheaves on S1 . Determine the torsion elements of this group when you replace Q with Q. Show that if we replace S1 with any connected

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variety and Q with C, then we obtain the structure of a complex Lie group (a.k.a. the character variety for rank one complex representations; one can define it for arbitrary rank, but needs geometric invariant theory to do so). Exercise 1.7.5. (Locally constant sheaves and representations of the fundamental group) A locally constant sheaf (a.k.a. a local system) L on Y gives rise to a representation ρL : π1 (Y, y) → GL(Ly ): pick a loop γ(t) at y and use local trivializations of L along the loop to move vectors in Ly along Lγ(t) , back to Ly . Exercise 1.7.6. (Representations of the fundamental group and locally constant sheaves) Given a representation ρ : π1 (Y, y) → GL(V) into a finite dimensional y ) → (Y, y), build the quotient space vector space, consider a universal cover (Y,  (V × Y)/π1 (Y, y), take the natural map (projection) to Y and take the sheaf of its local sections. Show that this is a locally constant sheaf whose associated representation is ρ. Exercise 1.7.7. (Zeroth cohomology of a local system) Let X be a connected space. Let L be a local system on X, and write M for the associated π1 (X)-representation. Show that H0 (X; L) = Mπ1 (X) , where the right hand side is the fixed part of M under the π1 (X)-action. Exercise 1.7.8. (Cohomology of local systems on a circle) Fix an orientation of S1 and the generator T ∈ π1 (S1 ) that comes with it. Let L be a local system on S1 with associated monodromy representation M. Show that H0 (S1 , L) = ker((T − id) : M → M),

H1 (S1 , L) = coker((T − id) : M → M),

and that H>1 (S1 , L) = 0. (Hint: one way to proceed is to use Cech cohomology. Alternatively, embed S1 as the boundary of a disk and use relative cohomology (or dualize and use compactly supported cohomology, where the orientation is easier to get a handle on)). Exercise 1.7.9. (Fiber bundles over a circle: the Wang sequence) This is an extension of the previous exercise. Let f : E → S1 be a locally trivial fibration with fibre F and monodromy isomorphism T : F → F. Show that the Leray spectral sequence gives rise to a short exact sequence 0 → H1 (S1 , Rq−1 f∗ Q) → Hq (E, Q) → H0 (S1 , Rq f∗ Q) → 0. Use the previous exercise to put these together into a long exact sequence . . . → Hq (E, Q) −→ Hq (F, Q) −→ Hq (F, Q) → Hq+1 (E) → . . . where the middle map Hq (F; Q) → Hq (F; Q) is given by T ∗ − id. Make a connection with the theory of nearby cycles (which are not discussed in these notes). Exercise 1.7.10. (The abelian category Loc(Y)) Show that the abelian category Loc (Y) of locally constant sheaves of finite rank on Y is equivalent to the abelian

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category of finite dimensional π1 (Y, y)-representations. Show that both categories are noetherian (acc ok!), artinian (dcc ok!) and have a duality anti-selfequivalence. Exercise 1.7.11 below is in striking contrast with the category Loc, but also with the one of perverse sheaves, which admits, by its very definition, the antiself-equivalence given by Verdier duality. Exercise 1.7.11. (The abelian category Shc (Y) is not artinian.) Show that in the presence of such an anti-self-equivalence, noetherian is equivalent to artinian. Observe that the category Shc (Y) whose objects are the constructible sheaves (i.e.  there is a finite partition of Y = Yi into locally closed subvarieties to which the sheaf restricts to a locally constant one (always assumed to be of finite rank!) is abelian and noetherian, but it is not artinian. Deduce that Shc (Y) does not admit an anti-self-equivalence. Give an explicit example of the failure of dcc in Shc (Y). Prove that Shc (Y) is artinian if and only if dim Y = 0. Exercise 1.7.12. (Cyclic coverings) Show that the direct image sheaf sheaf R0 f∗ Q for the map S1 → S1 , t → tn is a semi-simple locally constant sheaf of rank n; find its simple summands (one of them is the constant sheaf QS1 and the resulting splitting is given by the trace map). Do the same for R0 f∗ Q. Exercise 1.7.13. (Indecomposable non simple) The rank two locally constant sheaf on S1 given by the non-trivial unipotent 2 × 2 Jordan block is indecomposable and is neither simple nor semi-simple. Make a connection between this locally constant sheaf and the Picard-Lefschetz formula for the degeneration of a curve of genus one to a nodal curve. Amusing monodromy dichotomy. There is an important and amusing dichotomy concerning local systems in algebraic geometry (which we state informally): the global local systems arising in complex algebraic geometry via the decomposition theorem are semi-simple (i.e. completely reducible; related to the Zariski closure of the image of the fundamental group in the general linear group being reductive); the restriction of these local systems to small punctured disks with centers at infinity (degenerations), are quasi-unipotent, i.e. unipotent after taking a finite cyclic covering if necessary. This local quasi-unipotency is in some sense the opposite of the global complete reducibility. Quick review of Deligne’s 1972 and 1974 theory of mixed Hodge structures [25–27]; see also [31]. Deligne discovered the existence of a remarkable structure, a mixed Hodge structure, on the singular cohomology of a complex algebraic variety X (not necessarily smooth, complete or irreducible): there is an increasing filtration Wk H∗ (X, Q) and a decreasing filtration Fp H∗ (X, C) (with conjugate filtrapq ∗ tion denoted by F) such that the graded quotients GrW k H (X, C) = ⊕p+q=k Hk , where the splitting is induced by the (conjugate and opposite) filtrations F, F; i.e. (W, F, F) induce pure Hodge structure of weight k on GrW k . This structure is

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Perverse sheaves and the topology of algebraic varieties

canonical and functorial for maps of complex algebraic varieties. Important: one has that a map of mixed Hodge structures f : A → B is automatically strict, i.e. if f(a) ∈ Wk B, then there is a  ∈ Wk A with f(a  ) = f(a). Kernels, images and cokernels of pull-back maps in cohomology inherit such a structure. If X is a projective manifold, we get the known Hodge (p, q)-decomposition: Hi (X, C) = ⊕p+q=i Hpq (X). It is important to take note that for each fixed i pq i we have Hi (X, C) = ⊕k GrW k H (X, C) = ⊕k ⊕p+q=k Hk (X) which may admit several non zero k summands for k = i. In this case, we say that the mixed Hodge structure is mixed. This happens for the projective nodal cubic: H1 = H0,0 0 , . Here are some “inequalities” for the and for the punctured affine line H1 = H1,1 2 d d / [0, 2d]; if X is complete, then GrW weight filtration: GrW k H = 0 for k ∈ k>d H = 0; W d d if X is nonsingular, then Grk 0, R j∗ QU is skyscraper at o with stalk H (U, Q). Compute Hq c (Y, Q). Give a necessary and sufficient condition on the cohomology of V that ∼ H2d+2−q (Y, Q). Observe ensures that Y satisfies Poincaré duality Hq (Y, Q) = c that if V is a curve this condition boils down to it having genus zero. Remark: once you know about a bit about Verdier duality, this exercise tells you that the complex QY [dim Y] is Verdier self dual if and only if V meets the condition you have identified above; in particular, it does not if V is a curve of positive genus.

Fact 1.7.17. (ICY , Y a cone over a projective manifold V) Let things be as in Exercise 1.7.16. By adopting the definition of the intersection complex as an iterated push-forward followed by truncations, as originally given by GoreskyMacPherson, the intersection complex of Y is defined to be ICY := τd Rj∗ QU , where we are truncating the image direct complex Rj∗ QU in the following way: keep the same entries up to degree d − 1, replace the d-th entry by the kernel of the differential exiting it and setting the remaining entries to be zero; the resulting cohomology sheaves are the same as the ones for Rj∗ QU up to degree d included, and they are zero afterwards. More precisely, the cohomology sheaves of this / [0, d], H0 = QY , and for 1  q  d, Hi complex are as follows: Hq = 0 for q ∈ q is skyscaper at o with stalk Hprim (V, Q). Here is a justification for this definition: while Q[dim Y] usually fails to be Verdier self-dual, one can verify directly that the intersection complex ICY := ICY [dim Y] is Verdier self-dual. In general, if we were to truncate at any other spot, then we would not get this self-duality behavior (unless we truncate at minus 1 and get zero). Note that the knowledge of the cohomology sheaves of a complex, e.g. ICY , is important information, but it does not characterize the complex up to isomorphism. ∼ ⊕Rq f∗ QX [−q]) Let Exercise 1.7.18. (Example of no decomposition Rf∗ QX = things be as in Exercise 1.7.16 and assume that V is a curve of positive genus, so that Y is a surface. Let f : X → Y be the resolution obtained by blowing up the vertex o ∈ Y. Use the failure of the self-duality of QY [2] to deduce that QY is not a direct summand of Rf∗ QX . Deduce that Rf∗ QX  ⊕Rq f∗ QX [−q]. (The reader is invited to check [17, §3.1] out: it contains an explicit computation dealing with this example showing that as you try split QY off Rf∗ QX , you meet an obstruction; instead, you end up splitting ICY off Rf∗ QX , provided you have defined ICY as the truncated push-forward as above.) Fact 1.7.19. (Intersection complexes on curves) Let Y o be a nonsingular curve and L be a locally constant sheaf on it. Let j : Y o → Y be an open immersion into another curve (e.g. a compactification). Then ICY (L) = j∗ L[1] (definition of IC via push-forward/truncation). Note that if y ∈ Y is a nonsingular point, then the stalk (j∗ L)y is given by the local monodromy invariants of L around a small loop about y. The complex Rj∗ L[1] may fail to be Verdier self-dual, whereas its truncation τ−1 Rj∗ L[1] = j∗ L[1] = ICY (L) is Verdier self-dual. Note that we have

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a factorization Rj! L[1] → ICY (L) → Rj∗ L[1]. This is not an accident; see the end of §2.6. Exercise 1.7.20. (Blow-ups) Compute the direct image sheaves Rq f∗ Q for the blowup of Cm ⊆ Cn (start with m = 0; observe that there is a product decomposition of the situation that allows you to reduce to the case m = 0). Same question for the composition of the blow up of C1 ⊆ C3 , followed by the blowing up of a positive dimensional fiber of the first blow up. Observe that in all ∼ ⊕Rq f∗ Q[−q]. Guess the shape of cases, one gets an the decomposition Rf∗ Q = the decomposition theorem in both cases. Exercise 1.7.21. (Examples of the decomposition theorem) Guess the exact form of the cohomological and “derived” decomposition theorem in the following cases: 1) the normalization of a cubic curve with a node and of a cubic curve with a cusp; 2) the blowing up of a smooth subvariety of an algebraic manifold; 3) compositions of various iterations of blowing ups of nonsingular varieties along smooth centers; 4) a projection F × Y → Y; 5) the blowing up of the vertex of the affine cone over the nonsingular quadric in P3 ; 6) same but for the projective cone; 7) blow up the same affine and projective cones but along a plane through the vertex of the cone; 8) the blowing up of the vertex of the affine/projective cone over an embedded projective manifold. Exercise 1.7.22. (decomposition theorem for Lefschetz pencils) Guess the shape  → P1 on a nonsingular of the decomposition theorem for a Lefschetz pencil f : X projective surface X. Work out explicitly the invariant cycle theorems in this case. Do the same for a nonsingular projective manifold. When do we get skyscraper contributions? Exercise 1.7.23. (Künneth for the derived image complex) One needs a little bit of working experience with the derived category to carry out what follows below. But try anyway. Let f : X := Y × F → Y. A class aq ∈ Hq (X, Q) is the same thing as a map in the derived category aq : QX → QX [q]. First pushing forward via Rf∗ , then observing that Rf∗ f∗ QY = Rf∗ QX , and, finally, pre-composing with the adjunction map QY → Rf∗ QX , yields a map aq : QY → Rf∗ QX [q]. Take aq to be of the form pr∗F αq . Obtain a map αq : Hq (F) → Rf∗ QX [q]. Next, shift this map to get αq : Hq (F)[−q] → Rf∗ QX . Show that the map induces the “identity” on the q-th direct image sheaf and zero on the other direct image sheaves. Deduce that  q q αq : ⊕q H (F)[−q] → Rf∗ Q X is an isomorphism in the derived category inducing the “identity” on the cohomology sheaves. Observe that you did not make any choice in what above, i.e. the resulting isomorphism is canonical, whereas in Deligne’s theorem 1.5.3, one does not obtain a canonical isomorphism. Exercise 1.7.24. (Deligne’s theorem as a special case of the decomposition theorem) Keeping in mind that if So = S, then ICS (L) = L, recover the Deligne theorem from the decomposition theorem.

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. Lecture 2: The category of perverse sheaves P(Y) Summary of Lecture 2. The constructible derived category. Definition of perverse sheaves. Artin vanishing and its relation to a proof of the Lefschetz hyperplane theorem for perverse sheaves. The perverse t-structure (really, only the perverse cohomology functors!). Beilinson’s and Nori’s equivalence theorems. Several equivalent definitions of intersection complexes. 2.1. Three "Whys", and a brief history of perverse sheaves Why intersection cohomology? Let us look at (1.3.2) for X and Y nonsingular: ∼ ⊕q,EV IH∗−q (S, L), H∗ (X, Q) = q i.e. the l.h.s. is ordinary cohomology, but the r.h.s. is not any kind of ordinary cohomology on Y: we need intersection cohomology to state the decomposition theorem, even when X and Y are nonsingular. The intersection cohomology groups of a projective variety enjoy a battery of wonderful properties (Poincaré-HodgeLefschetz package). In some sense, intersection cohomology nicely replaces singular cohomology on singular varieties, but with a funny twist: singular cohomology is functorial, but has no Poincaré duality; intersection cohomology has Poincaré duality, but is not functorial! Why the constructible derived category? The cohomological Deligne theorem (1.1.4) for smooth projective maps is a purely cohomological statement and it can be proved via purely cohomological methods (hard Lefschetz + Leray spectral sequence). The cohomological decomposition theorem (1.3.2) is also a cohomological statement. However, there is no known proof of this statement that does not make use of the formalism of the middle perversity t-structure present in the constructible derived category: one proves the derived version (1.6.3) and then deduces the cohomological one (1.3.2) by taking cohomology. Actually, the definition of perverse sheaves does not make sense if we take the whole derived category, we need to take complexes with cohomology sheaves supported at closed subvarieties (not just classically closed subsets). We thus restrict to an agreeable, yet flexible, class of complexes: the “constructible complexes”. Why perverse sheaves? Intersection complexes, i.e. the objects appearing on both sides of the decomposition theorem (1.6.3) are very special perverse sheaves. In fact, in a precise way, they form the building blocks of the category of perverse sheaves: every perverse sheaf is an iterated extension of a collection of intersection complexes. Perverse sheaves satisfy their own set of beautiful properties: Artin vanishing theorem, Lefschetz hyperplane theorem, stability via duality, stability via vanishing and nearby cycle functors. As mentioned above, the known proofs of the decomposition theorem use the machinery of perverse sheaves. A brief history of perverse sheaves. Intersection complexes were invented by Goresky-MacPherson as a tool to systematize, strengthen and widen the scope

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Perverse sheaves and the topology of algebraic varieties

of their own intersection cohomology theory. For example, their original geometric proof of Poincaré duality can be replaced by the self-duality property of the intersection complex. See also S. Kleiman’s very entertaining survey [38]. The conditions leading to the definition of perverse sheaves appeared first in connection with the Riemann-Hilbert correspondence established by Kashiwara and by Mekbouth: their result is an equivalence of categories between the constructible derived category (which we have been procrastinating to define) and the derived category of regular holonomic D-modules (which we shall not define); the standard t-structures, given by the standard truncations met in Exercise 1.7.17, of these two categories do not correspond to each other under the Riemann-Hilbert equivalence; the conditions leading to the “conditions of support” defining of perverse sheaves are the (non-trivial) translation in the constructible derived category of the conditions on the D-module side stating that a complex of D-modules has trivial cohomology D-modules in positive degree. It is a seemingly unrelated, yet remarkable and beautiful fact, that the conditions of support so obtained are precisely what makes the Artin vanishing Theorem 2.4.1 work on an affine variety. As mentioned above, Gelfand and MacPherson conjectured the decomposition theorem for Rf∗ ICX . Meanwhile, Deligne had developed a theory of pure complexes for varieties defined over finite fields and established the invariance of purity under push-forward by proper maps. Gabber proved that the intersection complex of a pure local system, in that context, is pure. The four authors of [2] introduced and developed systematically the basis for the theory of t-structures, especially with respect to the middle perversity. They then proved that the notions of purity and perverse t-structure are compatible: a pure complex splits over the algebraic closure of the finite field as prescribed by the r.h.s. of (1.6.3). The decomposition theorem over the algebraic closure of a finite field follows when considering the purity result for the proper direct image mentioned above. The whole of Ch. 6 in [2], aptly named “De F à C”, is devoted to explaining how these kind of results over the algebraic closure of a finite field yield results over the field of complex numbers. This established the original proof of the decomposition theorem over the complex numbers for semi-simple complexes of geometric origin (see [2, 6.2.4, 6.2.5]), such as ICX . M. Saito has developed in [48] the theory of mixed Hodge modules which yields the desired decomposition theorem when M underlies a variation of polarizable pure Hodge structures. M. A. de Cataldo and L. Migliorini have given a proof based on classical Hodge theory of the decomposition theorem when M is constant [16]. Finally, the decomposition theorem stated in (1.6.3) is the most general statement currently available over the complex numbers and is due to work of C. Sabbah [47] and T. Mochizuki [41, 42] (where this is done in the essential case of projective maps of quasi projective manifolds; it is possible to extend it to proper

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maps of algebraic varieties). The methods (tame harmonic bundles, D-modules) are quite different from the ones discussed in these lectures. 2.2. The constructible derived category D(Y) The decomposition theorem isomorphisms (1.6.3) take place in the “constructible derived category” D(Y). It is probably a good time to try and give an idea what this category is. Constructible sheaf. A sheaf F on Y is constructible if there is a finite disjoint  union decomposition Y = a Sa into locally closed subvarieties such that the restriction F|Sa are locally constant sheaves of finite rank. This is a good time to look at Exercise 2.7.1. Constructible complex. A complex C of sheaves of rational vector spaces on Y is said to be constructible if it is bounded (all but finitely many of its cohomology sheaves are zero) and its cohomology sheaves are constructible sheaves. See the most-important Fact 2.2.1. Constructible derived category. The definition of D(Y) is kind of a mouthful: it is the full subcategory of the derived category D(Sh(Y, Q)) of the category of sheaves of rational vector spaces whose objects are the constructible complexes. It usually takes time to absorb these notions and to absorb the apparatus it gives rise to. We take a different approach and we try to isolate some of the aspects of the theory that are more relevant to the decomposition theorem. We do not dwell on technical details. Cohomology. Of course, the first functors to consider are cohomology and cohomology with compact supports Hi (Y, −), Hic (Y, −) : D(Y) → D(point). They can be seen as special cases of derived direct images. Derived direct images Rf∗ , Rf! . We can define derived direct image maps Rf∗ , Rf! : D(X) → D(Y), for every map f : X → Y. The first thing to know is that H∗ (X, C) = H∗ (Y, Rf∗ C) and that H∗c (X, C) = H∗c (Y, Rf! C), so that we may view them as generalizing cohomology. Pull-backs. The pull-back functor f∗ is probably the most intuitive one. The extraordinary pull-back functor f! is tricky and we will not dwell on it. It is the right adjoint to Rf! ; for open immersions, f! = f∗ ; for closed immersions it is the derived version of the sheaf of sections supported on the closed subvariety; for smooth maps of relative dimension d, f! = f∗ [2d]. A down-to-earth reference for f! and duality I like is [35] (good also, among other things, as an introduction to Borel-Moore homology). I also like [32]. There is also the seemingly inescapable, and nearly encyclopedic [37]. Fact 2.2.1. A good reference is [4]. Given C ∈ D(Y), and y ∈ Y there is a system of “standard neighborhoods” Uy () (think of 0 <   1 as the radius of an Euclidean ball; of course, our Uy () are singular, if Y is singular at y) such that H∗ (Uy (), C) and H∗c (Uy (), C) are “constant” (make the meaning of constant precise) for 0 <   1. The Uy () are cofinal in the system of neighborhoods of

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y. We have a canonical identification H∗ (C)y = H∗ (Uy (), C) for 0 <   1. This is very important for many reasons. Let us give one. Let f : X → Y be a map and let K ∈ D(X). Since Rf∗ K is constructible, we have (R∗ f∗ K)y = H∗ (f−1 (Uy (), K) for 0 <   1. Caution: there is a natural map to H∗ (f−1 (y), K), but this map is in general neither surjective, nor injective; if the map is proper, then it is an isomorphism (proper base change). ∼ D(Y) whose Verdier duality. This is an anti-self equivalence (−)∨ : D(Y)op = ∗ ∨ −∗ defining property is a natural perfect pairing H (Y, C ) × Hc (Y, C) −→ Q, or, equivalently, of a canonical isomorphism (2.2.2)

∼ H−∗ (Y, C)∨ . H∗ (Y, C∨ ) = c

If Y is nonsingular irreducible, then Q∨ Y = Q Y [2 dim Y], and we get an identifica∨ , i.e. Poincaré duality. (Y, Q) tion H∗+2 dim Y (Y, Q) = H−∗ c Stability of constructibility. It is by no means obvious, nor easy, that if C is constructible, then Rf∗ C is constructible. This can be deduced from the Thom isotopy lemmas5 [33]. This is a manifestation of the important principle that constructibility is preserved under all the “usual” operations on the derived category of sheaves on Y (see [4]). The list above is more complete once we include the derived RHom, the tensor product of complexes (it is automatically derived when using Q-coefficients), the nearby and vanishing cycle functors, etc. Duality exchanges. Verdier duality exchanges Rf∗ with Rf! , and f∗ with f! , i.e., Rf∗ (C∨ ) = (Rf! C)∨ and f∗ (K∨ ) = (f! K)∨ . Here is a nice consequence: let f be proper (so that Rf∗ = Rf! ), then if C is self-dual, then so is Rf∗ C. The importance of being proper. Proper maps are important for many reasons: Rf! = Rf∗ ; the duality exchanges simplify; the proper base change theorem holds, a special case of which tells us that (Rq f∗ QX )y = Hq (f−1 (y), Q) (see Exercise 1.7.1); Rf∗ preserves pure complexes (Frobenius, mixed Hodge modules); Grothendieck trace formula is about Rf! ; the decomposition theorem is about proper maps. Adjunctions. We have adjoint pairs (f∗ , Rf∗ ) and (Rf! , f! ), hence natural transformations: Id → Rf∗ f∗ and Rf! f! → Id. By applying cohomology to the first one, we get the pull-back map in cohomology, and by applying cohomology with compact supports to the second, we obtain the push-forward in cohomology with compact supports. The functors RHom and ⊗ also form an adjoint pair (you should formalize this). The attaching triangles and the long exact sequences we already know. By combining adjunction maps, we get some familiar situations from algebraic topology. Let j : U → Y ← Z : i be a complementary pair of open/closed embeddings. 5 The main two points are: 1) given a map f : X → Y of complex algebraic varieties, there is a dis joint union decomposition Y = i Yi into locally closed subvarieties such that Xi := f−1 Yi → Yi is a topological fiber bundle for the classical topology; 2) algebraic maps can be completed compatibly with the previous assertion (you may want to sit down and come up with a reasonable precise statement yourself).

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Given C ∈ D(Y), we have the distinguished triangle i∗ i! C → C → j∗ j∗ C → i! i! C[1] and, by applying sheaf cohomology, we get the long exact sequence of relative cohomology . . . → Hq (Y, U, C) → Hq (Y, C) → Hq (U, C|U ) → Hq+1 (Y, U, C) → . . . If g : Y → Z is a map, then we can push forward the attaching (distinguished) triangle and obtain a distinguished triangle, which will also give rise to hosts of long exact sequences when fed to cohomological functors. By dualizing, we get j! j∗ C → C → i∗ i∗ C → j! j∗ C[1] and by taking cohomology with compact supports, we get the long exact sequence of cohomology with compact supports q q q+1 (U, C) → . . . . . . → Hq c (U, C) → Hc (Y, C) → Hc (Z, C|Z ) → Hc

This is nice, and used very often, because it gives the usual nice relation between the compactly supported Betti numbers of the three varieties (U, Y, Z). Exercise 2.7.11 asks you to use the first attaching triangle and its push-forward when studying the resolution of singularities of a germ of an isolated surface singularity. This is an important example: it shows how the intersection complex of the singular surface arises; it relates the non degeneracy of the intersection form on the curves contracted by the resolution, to the decomposition theorem for the resolution map. The careful study of this example allows to extract many general ideas and patterns, specifically how to relate the non degeneracy of certain local intersection forms to a proof of the decomposition theorem. 2.3. Definition of perverse sheaves Before we define perverse sheaves. The category of perverse sheaves on an algebraic variety is abelian, noetherian, anti-self-equivalent under Verdier duality, artinian. The cohomology groups of perverse sheaves satisfy Poincaré duality, Artin vanishing and the Lefschetz hyperplane theorem. They are stable under the nearby and vanishing cycle functors. The simple perverse sheaves, i.e. the intersection complexes with simple coefficients, satisfy the decomposition theorem and the relative hard Lefschetz theorem, and their cohomology groups satisfy the Hard Lefschetz theorem and, when the coefficients are “Hodge-theoretic”, the Hodge-Riemann bilinear relations. Perverse sheaves play important roles in the topology of algebraic varieties, arithmetic algebraic geometry, singularity theory, combinatorics, representation theory, geometric Langlands program. Even though not everyone agrees (e.g. I), one may say that perverse sheaves are more natural than constructible sheaves; see Remark 2.4.3. Perverse sheaves on singular varieties are close in spirit to locally constant sheaves on algebraic manifolds. Perverse sheaves have at least one drawback: they are not sheaves!6 6 The “sheaves” in perverse sheaves is because the objects and the arrows in P(Y) can be glued from local data, exactly like sheaves; this is false for the derived category and for D(Y)!; do you know why? As to the term “perverse”, see M. Goresky’s post on Math Overflow: What is the etymology of the term perverse sheaf?

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Perverse sheaves and the topology of algebraic varieties

The conditions of support and of co-support. We say that a constructible complex C ∈ D(Y) satisfies the conditions of support if dim supp Hi (C)  −i, for every i ∈ Z, and that it satisfies the conditions of co-support if its Verdier dual C∨ satisfies the conditions of support. Definition 2.3.1. (Definition of the category P(Y) of perverse sheaves) We say that P ∈ D(Y) is a perverse sheaf if P satisfies the conditions of support and of co-support. The category P(Y) of perverse sheaves on Y is the full subcategory of the constructible derived category D(Y) with objects the perverse sheaves. Conditions of (co)support and vectors of dimensions. To fix ideas, suppose that dim Y = 4. The following vector exemplifies the upper bounds for the dimensions of the supports of the cohomology sheaves Hi for −4  i  0 (outside of this interval, the cohomology sheaves of a perverse sheaf can be shown to be zero) (4, 3, 2, 1, 0). For comparison, the analogous vector for an intersection complex of the form ICY (L) is (4, 2, 1, 0, 0). For a table giving a good visual for the conditions of support and co-support for perverse sheaves and for intersection complexes, see [19, p.556]. It is important to keep in mind that the support conditions are conditions on the stalks (direct limits of cohomology over neighborhoods) of the cohomology sheaves, whereas the conditions of co-support are conditions on the co-stalk (inverse limits of cohomology with compact supports over neighborhoods) and as such are maybe a bit less intuitive; see Exercise 1.7.16, where this issue is tackled for the constant sheaf on the affine cone over an embedded projective manifold. On the other hand, if for some reason we know that complex C ∈ D(Y) is Verdier self-dual, then it is perverse if and only if it satisfies the conditions of support. Note that the derived direct image via a proper map of a self-dual complex is self-dual. This simple remark is very helpful in practice. This is a good time to carry out Exercises 2.7.3 and 2.7.4. 2.4. Artin vanishing and Lefschetz hyperplane theorems Conditions of support and Artin vanishing theorem. The conditions of support seem to have first appeared in the proof of the Artin vanishing theorem for constructible sheaves in SGA 4.3.XIV, Théorème 3.1, p.159: let Y be affine and F ∈ Shc (Y); then H∗ (Y, F) = 0 for ∗ > dim Y. Note that F[dim Y] satisfies the conditions of support, and that Artin’s wonderful proof works for a constructible complex satisfying the conditions of support. Note also that if Y is nonsingular and F = QY , the result is affordable by means of Morse theory7 ! Since perverse sheaves satisfy the conditions of support and co-support, we see that they automatically satisfy the “improved” version of the Artin vanishing theorem. 7 The

singular case is one of the reasons for the existence of the book [33].

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Theorem 2.4.1. (Artin vanishing theorem for perverse sheaves)8 Let P be a perverse sheaf on the affine variety Y. Then H∗ (Y, P) = 0, for ∗ not in [− dim Y, 0] and H∗c (Y, P) = 0, for ∗ not in [0, dim Y]. “Proof.” See Exercise 2.7.5. Exactly as in the Morse-theory proof of the weak Lefschetz theorem, the improved Artin vanishing theorem implies Theorem 2.4.2. (Lefschetz hyperplane theorem for perverse sheaves). Let Y be a quasi projective variety, let P ∈ P(Y) and let Y1 ⊆ Y be a general hyperplane section. Then H∗ (Y, P) → H∗ (Y1 , P|Y1 ) is an isomorphism for ∗  −2, and is injective for ∗ = −1. There is a similar statement for compactly supported cohomology (guess it!). Proof. We give the proof when Y is projective. Let j : U := Y \ Y1 → Y ← Y1 : i be the natural maps. We have the attaching triangle Rj! j∗ P → P → Ri∗ i∗ P → and the long exact sequence of cohomology (= cohomology with compact supports because Y is projective!) ∗ −k (Y, P) → H−k (Y1 , P|Y1 ) → H−k+1 (Y, Rj! j∗ P) → . . . . . . → H−k c (Y, Rj! j P) → H ∗ −k ∗ We need to show that H−k c (Y, Rj! j P) = Hc (U, j P) = 0 for −k < 0, but this is Artin vanishing for perverse sheaves on the affine U. 

Remark 2.4.3. (Lefschetz hyperplane theorem: Perverse sheaves vs. sheaves) As the proof given above shows, once we assume the projectivity of Y, any hyperplane section Y1 will do. This is similar to the classical proof of the Lefschetz hyperplane theorem due to Andreotti-Frankel (following a suggestion by Thom) and contained in Milnor’s Morse Theory (jewel) book [40], where, if Y is projective, then we only need to pick a hyperplane section that contains all the singularities of Y, so that the desired vanishing stems from Lefschetz duality and from Morse theory. This shows that even the constant sheaf is not well-behaved on singular spaces! If we try and repeat the proof above for the constant sheaf on a singular space, we stumble into the realization that we do not have the necessary Artin vanishing for cohomology with compact supports for the constant sheaf on the possibly singular U. This issue disappears if we use perverse sheaves! The simple perverse sheaves are the intersection complexes. Since the category P(Y) is artinian, every perverse sheaf P ∈ P(Y) admits an increasing finite filtration with quotients simple perverse sheaves. The important fact is that the simple perverse sheaves are the intersection complexes ICS (L) seen above with L simple! (Exercise 2.7.10). There is a shift involved: ICS (L) is not perverse on the nose, but if we set (2.4.4)

ICS (L) := ICS (L)[dim S],

8 Note that there is no sheaf analogue of Artin vanishing for compactly supported cohomology: the Verdier dual of a constructible sheaf is not a sheaf!

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Perverse sheaves and the topology of algebraic varieties

then the result is a perverse sheaf on Y. One views these complexes on the closed subvarieties i : S → Y as complexes on Y supported on S via i∗ (we do not do this to simplify the notation). There is an important collection of results in intersection cohomology that generalize to the singular setting the beloved collection of classical results that hold for complex algebraic manifolds. See Exercise 2.7.7. 2.5. The perverse t-structure The constructible derived category D(Y) comes equipped with the standard t-structure—i.e. the truncation functors are the standard ones—whose heart is the abelian category Shc (Y) ⊆ D(Y) of constructible sheaves. A t-structure on a triangulated category is an abstraction of the notion of standard truncation [2]. A triangulated category may carry several inequivalent t-structures. The middle perversity t-structure on D(Y). The category of perverse sheaves P(Y) is also the heart of a t-structure on D(Y), the middle-perversity t-structure. Instead of dwelling on the axioms, here is a short discussion. The perverse sheaf cohomology functors. Every t-structure on a triangulated category comes with its own cohomology functors; the standard one comes with the cohomology sheaf functors. The perverse t-structure then comes with the perverse cohomology sheaves pHi : D(Y) → P(Y) which are . . . cohomological, i.e. turn distinguished triangles into long exact sequences A → B → C → A[1] =⇒ . . . → pHi (A) → pHi (B) → pHi (C) → pHi+1 (A) → . . . , and, moreover, we have (2.5.1)

Hi (C[j]) = pHi+j (C).

p

Let us mention that C ∈ D(Y) satisfies the conditions of support if and only if its perverse cohomology sheaves are zero in positive degrees; similarly, for the conditions of co-support (swap positive with negative). Kernels, cokernels. Once you have the cohomology functors, you can verify that P(Y) is abelian: take an arrow a : P → Q in P(Y), form its cone C ∈ D(Y), and then you need to verify that pH−1 (C) → P is the kernel and that Q → pH0 (C) is the cokernel. What is the image? Verdier duality exchange: (2.5.2)

Hi (C∨ ) = pH−i (C)∨

p

∼ C∨ , then ( pHi (C))∨ = pH−i (C) and if, in addition, f is proper, then If C = (2.5.3)

∼ Rf∗ C. (Rf∗ C)∨ =

The perverse cohomology sheaves of a complex do not determine the complex. However, Exercise 4.7.8 tells us that in the decomposition theorem (1.6.3) we may write  p c ∼ H (Rf∗ ICX (M))[−c]. Rf∗ ICX (M) = c∈Z

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The perverse cohomology sheaf construction is a way to get perverse sheaves out of any complex. So there are plenty of perverse sheaves. In fact, we have the following two rather deep and very (!) surprising facts: Theorem 2.5.4. (The constructible derived category as a derived category) Let Y be a variety. (1) ([46]9 ) D(Y, Z) with its standard t-structure is equivalent to Db (Shc (Y, Z)) with its standard t-structure. (2) ([1]10 ) D(Y, Q)) with its perverse t-structure is equivalent to Db (P(Y, Q)) with its standard t-structure. Exercise 2.7.9 introduces another construction that leads to special perverse sheaves, i.e. the intermediate extension functor j!∗ . This is crucial, in view of the fact that the intersection cohomology complex ICS (L) can be defined as the intermediate extension of L[dim S] from the open subvariety So ⊆ Sreg ⊆ S on which L is defined to the whole of S, and thus to any variety that contains S as a closed subvariety. 2.6. Intersection complexes Recall the conditions of support and co-support for a complex P to be a complex of perverse sheaves: dim supp Hi (C)  −i, and the same for C∨ . The original definition of intersection complex ICS (L) of an enriched variety (where we start with L a locally constant sheaf on some non-empty So ⊆ Sreg ⊆ S) involves repeatedly pushing-forward and standard-truncating across the strata of a suitable stratification of S, starting from So ; see [2, Proposition 2.1.11]. It is a fact that shrinking So does not effect the end result (this is an excellent exercise). The end result can be characterized as follows. Conditions of (co)support for intersection complexes. The intersection complex ICS (L) of an enriched variety (S, L) is the complex C, unique up to unique isomorphism, subject to the conditions of support and co-support: C|So = L[dim S], dim supp Hi  −i − 1 for every i = − dim S, and the “same” for C∨ . Recall the two vectors exemplifying the conditions of support in dimension four for perverse sheaves (4, 3, 2, 1, 0), and for intersection complexes (4, 2, 1, 0, 0). Using this characterization, the reader should verify that direct images under finite maps preserve intersection complexes and that the same is true for small maps (see §). Another characterization of intersection complexes. The intersection complex ICS (L) is the unique perverse sheaf extending its own restriction to an open dense subvariety U ⊆ S so that the extension is “minimal” in the following sense: it has no non zero perverse subobject or quotient supported on the boundary S \ U11 . 9 Nori’s

paper contains a lovely proof of Artin vanishing in characteristic zero. also proves his wonderful Lemma 3.3, a strengthening of the Artin vanishing in arbitrary characteristic; Nori calls it the “Basic Lemma” in [46]. 11 It may have, however, a non zero subquotient supported on the boundary; the formation of this kind of minimal—a.k.a. intermediate extension—is not exact on the relevant abelian categories: it preserves injective and surjective maps, but it does not preserve exact sequences; see [19, p.562]. 10 Beilinson

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Perverse sheaves and the topology of algebraic varieties

Intersection complexes as intermediate extensions. Let j : U → S be a locally closed embedding. Let P ∈ P(U). Take the natural map (forget the supports) Rj! P → Rj∗ P. Take the map induced at the level of 0-th perverse cohomology sheaves: a : pH0 (Rj! P) → pH0 (Rj∗ P). Define the intermediate extension of P on U to Y by setting j!∗ P := Im a ∈ P(S). Let (S, L) be an enriched variety and let j : So → S; we apply the intermediate extension functor to P := L[dim S] ∈ P(U := So ) and we end up with ICS (L)! The same conclusion holds if we take any U and P := (ICS (L))|U ) (the intersection complex is the intermediate extension of its restriction to any dense open subvariety). Since P(Y) is Noetherian and closed under Verdier duality, it is artinian, so that the Jordan-Holder theorem holds. Now it is a good time to carry out Exercise 2.7.10, which shows how to produce an “explicit” Jordan-Holder decomposition for perverse sheaves. The method also makes it clear that the simple objects in P(Y) are the ICS (L) with S ⊆ Y an irreducible closed subvariety and with L a simple locally constant sheaf on some Zariski dense open subset of the regular part of S. 2.7. Exercises for Lecture 2 Exercise 2.7.1. (Some very non constructible sheaves) Provide a definition of analytic constructibility in the context of the analytic Zariski topology which is parallel to the one given in §2.2 by dropping the condition on the finiteness of the collection of locally closed subvarieties. Use the closed embedding i : C → A1 of the Cantor set into the complex affine line to show that the direct image sheaf i∗ QC is not analytically constructible. Exercise 2.7.2. (Skyscraper sheaves) Classify the sheaves of rational vector spaces on a variety which are both constructible and injective. Example 2.7.3. (First (non) examples) If Y is of pure dimension and F ∈ Shc (Y), then F[dim Y] satisfies the conditions of support. In general, F[dim Y] is not perverse as its Verdier dual may fail to satisfy the condition of support. For example, the Verdier dual of QY [dim Y] is the shifted dualizing complex ωY [− dim Y] and the singularities of Y dictate whether or not it satisfies the conditions of support; see [18, §4.3.5-7]. If Y is nonsingular of pure dimension, and L is locally constant, then L[dim Y] is perverse, for its Verdier dual is L∨ [dim Y]. Exercise 2.7.4. (Some perverse sheaves) The derived direct image of a perverse sheaf via a finite map is perverse. Give examples showing that the derived direct image via a quasi-finite map of a perverse sheaf may fail to be perverse. Let j : X := C∗ → C =: Y be the natural open embedding; show that the natural map Rj! QX [1] → Rj∗ QX [1] in D(Y) is, in fact, in P(Y); determine kernel, cokernel and image. Show that if we replace C∗ with Cn \ {0}, then the map above is not one of perverse sheaves. Show that if instead of removing the origin, we remove a finite configuration of hypersurfaces, then we get a map of perverse sheaves.

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Show more generally that an affine open immersion j is such that Rj! and Rj∗ preserve perverse sheaves (hint: push-forward and use freely the Stein (rather than the affine) version of the Artin vanishing theorem to verify the conditions of support on small balls centered at points on the hypersurfaces at the boundary). Show that the direct image Rf∗ QX [2] with f : X → Y a resolution of singularities of a surface is perverse. Let f : X2d → Y 2d be proper and birational, with X nonsingular and irreducible, let y ∈ Y and let f be an isomorphism over Y \ y; give an "if and only if" condition that ensures that Rf∗ Q[dim X] is perverse. Determine the pairs m  n such that the blowing up of f : X → Y of Cm ⊆ Cn =: Y is such that Rf∗ QX [n] is perverse. Exercise 2.7.5. (Artin vanishing: from constructible to perverse sheaves; cohomological dimension) Assume the Artin vanishing theorem for constructible sheaves and deduce the one for the cohomology of perverse sheaves by use of the Grothendieck spectral sequence Hp (Y, Hq (P)) =⇒ Hp+q (Y, P). (Hint: the supports of the cohomology sheaves are closed affine subvarieties.) Dualize the result to obtain the Artin vanishing theorem for the cohomology with compact support of a perverse sheaf. Use a suitable affine covering of a quasi-projective variety Y to show that the cohomology and cohomology with compact supports of a perverse sheaf live in the interval [− dim Y, + dim Y]. What about a non quasi projective Y? Exercise 2.7.6. (Intersection complex via push-forward and truncation) The original Goresky-MacPherson definition of intersection complex, suggested to them by Deligne, involves repeated push-forward and truncation across the strata of a Whitney-stratification. Let us take j : Cn \ Cm=0 =: U → Y := Cn . The formula reads ICY := τ−1 Rj∗ QU [n]. Verify that the result is QY [n]. Do this for other values of m and verify that you get QY [n], again. Take a complete flag of linear subspaces in Cn , apply the general formula given by iterated push-forward and truncation, and verify that you get QY [n]. What is your conclusion? Exercise 2.7.7. (Hodge-Lefschetz package for intersection cohomology) Guess the precise statements of the following results concerning intersection cohomology groups: Poincaré duality, Artin vanishing theorem, Lefschetz hyperplane theorem, existence of pure and mixed Hodge structures, hard Lefschetz theorem, primitive Lefschetz decomposition, Hodge-Riemann bilinear relations. Exercise 2.7.8. (Injective or surjective?) Let j : A1 \ {0} → A1 be the natural open embedding. Verify that: the natural map j! Q → j∗ Q in Shc (A1 ) is injective; the natural map j! [1]Q → j∗ Q[1] is surjective in P(A1 ). Exercise 2.7.9. (Perverse cohomology sheaves and the intermediate extension functor) For j : Cn \ {0} → Cn the natural open embedding, compute pHi (Rj! Q) and pHi (Rj∗ Q). Let j : U → X be an open embedding and let P ∈ P(U). Let a : Rj! P → Rj∗ P be the natural map. Show that the assignment defined by

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Perverse sheaves and the topology of algebraic varieties

P → j!∗ P := Im{ pH0 (Rj! P) → pH0 (Rj∗ P)} is functorial. This is the intermediate extension functor. Find an example showing that it is not exact (exact:= it sends short exact sequences in P(U) into ones in P(X)). (Hint: punctured disk and rank two unipotent and non-diagonal matrices). Compute j!∗ QU [dim U] when j is the embedding of a Zariski-dense open subset of a nonsingular and irreducible variety. Same for the embedding of affine cones over projective manifolds minus their vertex into the cone. Compute j!∗ L[1] where U = C∗ and L is a locally constant sheaf on U. Exercise 2.7.10. (Jordan-Holder for perverse sheaves) Recall our standing assumptions: varieties are not assumed to be irreducible, nor pure-dimensional. Let P ∈ P(Y). Find a non empty open nonsingular irreducible subvariety j : U ⊆ Y such that Q := j∗ P = L[dim U] for a locally constant sheaf on U. Produce the natural commutative diagram with a  epimorphic and a  monomorphic mm6 P QQQQQ QQQ mmm m m QQQ mm m QQQ m mmm ( a pH0 (Rj Q) / pH0 (Rj∗ Q) ! PPP nn6 PPPa  a nnnn PPP PPP nnn ( nnn ICU (L) := Ima  .

Deduce formally, from the fact that the image of P contains Ima  , that P contains a subobject P  together with a surjective map b : P  → ICU (L); you can even choose P  to be maximal with this property, but it does not matter in what follows. Deduce that we have a filtration Ker b ⊆ P  ⊆ P with P  /Ker b = ICU (L). Use noetherian induction to prove that we can refine this two-step filtration to a filtration with successive quotients of the form ICS (L). Each local system L appearing in this way admits a finite filtration with simple quotients. Refine further to obtain a finite increasing filtration of P with successive quotients of the form ICS (L) with L simple. (In the last step you need to use the fact that the intermediate extension functor, while not exact, preserves injective maps.) Exercise 2.7.11. (Attaching the vertex to a cone) Use attaching triangles and resulting long exact sequences of cohomology to study C = Rf∗ QX , where f : X → Y is the resolution of the cone (affine and projective) over a nonsingular embedded projective curve obtained by blowing up the vertex. (See [17].)

. Lecture 3: Semi-small maps Summary of Lecture 3. Definition of semi-small map. Hard Lefschetz and HodgeRiemann bilinear relations for semi-small maps. Special form of the decomposition theorem for semi-small maps. Hilbert schemes of points on smooth surfaces and the picture

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of Grojnowski-Nakajima. The endomorphism and correspondence algebras are isomorphic and semi-simple. Hint of a relation to the Springer picture. 3.1. Semi-small maps Semi-small maps are a very special class of maps, e.g. they are necessarily generically finite. On the other hand, the blowing up of point in C3 is not semi-small. Resolutions of singularities are very rarely semismall (except in dimension  2). It is remarkable that semi-small maps appear in important situations, e.g. holomorphic symplectic contractions, quiver varieties, moduli of bundles on surfaces, Springer resolutions, convolution on affine grassmannians, standard resolutions of theta divisors and Hilbert-Chow maps for Hilbert schemes of points on surfaces. References include [14] and the beautiful book [6]. We now discuss some of their features. To simplify the discussion, we work with proper surjective maps f : X → Y with X nonsingular and irreducible. We start with what is likely to be the quickest possible definition of semi-small map. Definition 3.1.1. (Definition of semi-small map) The map f is said to be semismall if dim X ×Y X = dim X. Quick is good, but not always transparent. The standard definition involves consideration of the dimension of the locally closed loci Sk ⊆ Y where the fibers of the map have fixed dimension k. Then semi-smallness is the requirement that dim Sk + 2k  dim X for every k  0; see Exercise 3.6.1. Small maps. We say that the map f is small if it is semi-small and X ×Y X has a unique irreducible component of maximal dimension dim X (which one?). For semi-small maps, this is equivalent to having dim Sk + 2k < dim X for every k > 0. The blowing ups of Cm ⊆ Cn , m  n − 2 have positive-dimensional fibers isomorphic to Pn−m−1 and are semi-small if and only if m = n − 2. None of these is small. The blowing up of the affine cone over the nonsingular quadric in P3 along a plane thru the vertex is a small map, with fiber over the vertex of the cone isomorphic to P1 . The blowing up of the vertex, which has fiber over the vertex isomorphic to the nonsingular quadric, is not. The Springer resolution of the nilpotent cone in a semi-simple Lie algebra is semi-small and Grothendieck-Springer simultaneous resolution is small. We shall meet both a bit later and show how they interact beautifully to give us a “decomposition theorem argument” for the presence of an action of the Weyl group of the Lie algebra on the cohomology of the fibers of the Springer resolution. The Weyl group does not act on the fibers! The following beautiful result of D. Kaledin is a source of many examples of highly non-trivial semi-small maps.

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Perverse sheaves and the topology of algebraic varieties

Theorem 3.1.2. (Holomorphic symplectic contractions are semi-small [36]) A projective birational map from a holomorphic symplectic12 nonsingular variety is semismall. It is amusing to realize that, for projective maps onto a projective target, semismallness and the Hard Lefschetz phenomenon are essentially equivalent. In fact, we have the following Theorem 3.1.3. (Hard Lefschetz for semi-small maps [14]) Let f : X → Y be a surjective projective map of projective varieties with X nonsingular and let η := f∗ L ∈ H2 (X, Q) be the first Chern class of the pull-back to X of an ample line bundle L on Y. The iterated cup product maps ηr : Hdim X−r (X, Q) → Hdim X+r (X, Q) are isomorphisms for every r  0 if and only if the map f is semi-small. In the semi-small case, we have the primitive Lefschetz decomposition and the Hodge-Riemann bilinear relations. Hodge-index theorem for semi-small maps. There is an important phenomenon concerning projective maps that is worth mentioning, i.e. the signature of certain local intersection forms [16]; for a discussion of these, see [17]. The situation is more transparent in the case of semi-small maps, where it is directly related to the Hodge-Riemann bilinear relations associated with Theorem 3.1.3. To have a clearer picture, let us limit ourselves to stating a simple, revealing and important special case. Let f : X → Y be a surjective semi-small projective map with X nonsingular of some even dimension 2d. Assume that f−1 (y) is ddimensional for some y ∈ Y. By intersecting in X, we obtain the refined symmetric intersection pairing H2d (f−1 (y)) × H2d (f−1 (y)) → Q, where we are intersecting the fundamental classes of the irreducible components of top-dimension d of this special fiber inside of X. The following is a generalization of a result of a famous result of Grauert’s for d = 1. Theorem 3.1.4. (Refined intersection forms have a precise sign) The refined intersection pairing above is (−1)d -positive-definite. By looking carefully at every proper map, similar refined intersection forms reveal themselves. One can prove that the decomposition theorem is (essentially) equivalent to the non-degeneracy of these refined intersection forms together with Deligne’s semi-simplicity of monodromy theorem; see [16]. Exercise 3.6.2 relates perverse sheaves and semi-small maps: if X in nonsingular, then f : X → Y is semi-small if and only if Rf∗ QX [dim X] is perverse. The decomposition theorem then tells us that  (3.1.5) f∗ QX [dim X] = pH0 (f∗ QX [dim X]) = ICS (L). (S,L)∈EV0

Question 3.1.6. What “are” the summands appearing in the decomposition theorem for semi-small maps? 12 An

even-dimensional variety with a closed holomorphic 2-form ω that is non-degenerate: ω is nowhere vanishing.

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3.2. The decomposition theorem for semi-small maps A little bit about stratifications. Even if not logically necessary, it simplifies matters to use the stratification theory of maps to clarify the picture a bit. There  is a finite disjoint union decomposition Y = a∈A Sa into locally closed nonsingular irreducible subvarieties Sa ⊆ Y such that f−1 (Sa ) → Sa is locally (for the classical topology) topologically a product over Sa . It is clear that this decomposition refines the one above given by the dimension of the fibers, so that dim Sa + 2 dim f−1 (s)  dim X for every s ∈ Sa . Since the map is assumed to be proper, it is also clear that all direct image sheaves Rq restrict to locally constant sheaves on every Sa . We call the Sa the strata (of a stratification of the map f). Definition 3.2.1. (Relevant stratum) We say that Sa is relevant if we have: dim Sa + 2 dim f−1 (s) = dim X. We denote by Arel ⊆ A the set of relevant strata. Exercise 3.6.3 shows that, for each relevant stratum Sa , the direct image sheaf Rdim X−dim Sa (cf. §1.2) restricted to Sa is locally constant, semi-simple, with finite monodromy. We denote this restriction by La . Theorem 3.2.2. (Decomposition theorem for semi-small maps) If f : X → Y is proper, surjective and semi-small with X nonsingular, then there is a direct sum decomposition  ICSa (La ). f∗ QX [dim X] = a∈Arel

Since the locally constant sheaf La is semi-simple, it admits the isotypical direct sum decomposition (1.5.5), i.e. we have La = ⊕χ La,χ ⊗ Ma,χ where χ ranges over a finite set of distinct isomorphism classes of simple locally constant sheaves on Sa and Ma,χ is a vector space of rank the multiplicity ma,χ of the locally constant sheaf Lχ in La . The decomposition theorem then reads  ICSa (La,χ ⊗ Ma,χ ). (3.2.3) f∗ QX [dim X] = a,χ

3.3. Hilbert schemes of points on surfaces and Heisenberg algebras An excellent reference is [43]. Let X be a nonsingular complex surface. For every n  0 we have the Hilbert scheme X[n] of n points on X. It is irreducible, nonsingular and of dimension 2n. There is a proper birational surjective map π : X[n] → X(n) onto the n-th symmetric product sending a length n zero dimensional subscheme of X to its support counting multiplicities. There is a natural stratification for the symmetric product of the map, which we now describe.  (n) First, we stratify the symmetric product variety X(n) = ν∈P(n) Xν , where P(n) is the set of partitions ν = {νj } of the integer n13 obtained by taking the locally closed, irreducible, nonsingular subvarieties of dimension 2l(ν) consisting 13 The

νj are a set of l(ν) positive integers adding up to n: l(ν) is called the length of ν.

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Perverse sheaves and the topology of algebraic varieties

 of points of the form i νi xi ∈ X(n) , with type {νi } given by ν. The remarkable (n) fact is that the fibers π−1 (xν ) of the points xν ∈ Xν are irreducible of dimen sion j (νj − 1) = n − l(ν). If we ignore the scheme structures, these fibers are isomorphic to products of punctual Hilbert schemes. It follows that the map π   (n) (n) is semi-small. In fact, X[n] = P(n) π−1 (Xν ) → P(n) Xν is the aforementioned stratification of the semi-small map π. All of its strata on X(n) are relevant. Since the fibers are all irreducible, the relevant locally constant sheaves are all constant of rank one. In particular, the decomposition theorem for π takes the form Rπ∗ QX[n] [2n] = ⊕ν∈P(n) IC (n) . Xν

(n)

A second remarkable fact is that the normalization of Xν can be identified with  (ai ) , where a is the number of a product of symmetric products X(ν) := n i i=1 X times that i appears in ν. This is a variety obtained by dividing a nonsingular variety by the action of a finite group; in particular, its intersection complex is the constant sheaf. By the IC normalization principle (Fact 4.5.3), we see that IC (n) is the push-forward of the shifted constant sheaf from the normalization. Xν Taking care of shifts, and waiting for the the dust to settle, we obtain the Göttsche formula: (recall that X(n) := {point})    H∗ (X[n] ) = H∗−cl(ν) (X(0) , Q), (3.3.1) H(X) := n0

n0 ν∈P(n)

where the colength is given by cl(ν) := n − l(ν). If we take X = C2 , then something remarkable emerges: look at (3.3.2) and (3.3.3). The formula above, taken for every n  0 gives (3.3.2)

∞ 

dim H∗ (C2

[n]

, Q) =

n=0

∞  j=1

1 . 1 − qj

Let R := Q[x1 , x2 , . . .] be the algebra of polynomials in the infinitely many indeterminates xi , declared to be of degree i. The infinite dimensional Heisenberg algebra H is the Lie algebra whose underlying rational vector space has basis {{di }i0 } and subject to the following relations: c0 is central, the di ’s commute with each other, the mi ’s commute with each other, and the commutator [di , mj ] = δ−i,j c0 . Then R is an irreducible H-module generated by 1 where di acts as formal derivation by xi and mj by multiplication by xj . The dimension an of the space of homogeneous polynomials of degree n is given by: (3.3.3)

∞  n=0

n

an q =

∞  j=1

1 . 1 − qj

Well, isn’t this a coincidence! The operators di , mi change the homogeneous degree of x-monomials by ±i. This, together with the formalism of correspondences in products, suggests that there should be geometrically meaningful cohomology

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classes in H∗ (C2 × C2 , Q) that reflect, on the Hilbert scheme side, the Heisenberg algebra action on the polynomial side. That this is indeed the case is due to Grojnowski and to Nakajima: they guessed what above, constructed algebraic cycles on the products of Hilbert schemes above that would be good candidates and then verified the Heisenberg Lie algebra relations. In fact, for every nonsingular surface X, there is an associated (Heisenberg-Clifford) algebra H(X) that acts geometrically and irreducibly on H(X) (3.3.1). [n]

[n±i]

3.4. The endomorphism algebra End(f∗ QX ) A reference here is [15]. Semi-simple algebras. A semi-simple algebra is an associative artinian (dcc) algebra over a field with trivial Jacobson ideal (the ideal killing all simple left modules). The Artin-Wedderburn theorem classifies the semi-simple algebras over a field as the ones which are finite Cartesian products of matrix algebras over finite dimensional division algebras over the field. Warm-up. Show that Md×d (Q) is semi-simple. Show that the upper triangular matrices do not form a semi-simple algebra. Hence if f := pr2 : P1 × P1 → P1 , then EndD(P1 ) (Rf∗ Q) is not a semi-simple algebra. Theorem 3.4.1. (Semi-small maps and semi-simplicity of the endomorphism algebra) In the setting of Theorem 3.2.2, the endomorphism Q-algebra EndD(Y) f∗ QX [dim X] is semi-simple. Proof. We see more important properties of intersection complexes at play, i.e. the Schur lemma phenomena for simple perverse sheaves. By simplicity, Hom(ICSa (Lχ ), ICSb (Lψ )) = δχ,ψ δa,b End(ICSa (Lχ )) (i.e., there are no non zero maps if they differ): in fact, look at kernel and cokernel and use simplicity. This leaves us with considering terms of the form End(ICSa (Lχ )) whose elements, for the same reason as above, are either zero, or are isomorphisms. These terms are thus division algebras Da,χ . It follows that  Mda,χ ×da,χ (Da,χ ), EndD(Y) (Rf∗ QX [dim X]) = a,χ

which is a semi-simple algebra by the Artin-Wedderburn theorem.



The endomorphism algebra as a geometric convolution algebra. A references here is also [6]. We can realize the algebra EndD(Y) (Rf∗ QX ) of endomorphisms in the derived category in geometric terms as the convolution algebra HBM 2 dim X (X ×Y X), which is thus semi-simple. Let us discuss this a bit.

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Perverse sheaves and the topology of algebraic varieties

Let X be a nonsingular projective variety. Then, we have an isomorphism of algebras between the first and last term (3.4.2)

End(H∗ (X)) =1 H∗ (X)∨ ⊗ H∗ (X) ∼ 2 H∗ (X) ⊗ H∗ (X) = ∼ 3 H∗ (X × X) = ∼ 4 H∗ (X × X) =

∼ 2 : Poincaré duality; = ∼ 3 : Künneth; = ∼ 4 : Poincaré where: =1 : linear algebra; = duality; and where: the algebra structure on the last term is the one given by the formalism of composition of correspondences in products (see Exercise 3.6.4). The classes Γ ∈ Hγ (X × X) appearing in Exercise 3.6.4 are called correspondences. This picture generalizes well, but not trivially, to proper maps f : X → Y from nonsingular varieties as follows. Theorem 3.4.3. (Correspondences and endomaps in the derived category) Let f : X → Y be a proper map from a nonsingular variety. There is a natural isomorphism ∼ HBM (X ×Y X) of Q-algebras. EndD(Y) (f∗ QX ) = 2 dim X For our semi-small maps, Exercise 3.6.7 provides an evident geometric basis of the vector space HBM 2 dim X (X ×Y X). Since there is a basis of HBM 2 dim X (X ×Y X) = EndD(Y) (Rf∗ Q X ) given by algebraic cycles, a formal linear algebra manipulation shows that if X is projective, then decomposition H∗ (X, Q) = ⊕a∈Arel IH∗−codim(Sa ) (Sa , La ) is compatible with the Hodge (p, q)-decomposition, i.e. it is given by pure Hodge substructures; see Exercise 3.6.8. In fact, one even has a canonical decomposition of Chow motives reflecting the decomposition theorem for semi-small maps; see [15]. Look at the related (deeper) Question 5.5.2. 3.5. Geometric realization of the representations of the Weyl group Excellent references here are [5, 6]. There is a well-developed theory of representations of finite groups G (character theory) into finite dimensional complex vector spaces. In a nearly tautological sense, this theory is equivalent to the representation theory of the group algebra C[G]. If we take the Weyl group W of any of the usual suspects, e.g. SLn (C) with Weyl group the symmetric group Sn , then we can ask whether we can realize the irreducible representations of W by using the fact that W is a Weyl group. Springer realized that this was indeed possible, and in geometric terms! In what follows, we do not reproduce this amazing story, but we limit ourselves to showing how the decomposition theorem14 allows us to introduce the action of the Weyl group on the cohomology of the Springer fibers. The Weyl group does not act algebraically on these fibers! Take the Lie algebra sln (C) of traceless n × n matrices. Inside of it there is the cone N with vertex the origin given by the nilpotent matrices. Take the flag variety  := {(n, f)|, n stabilizes f} ⊆ N × F. F, i.e. the space of complete flags f in Cn . Set N 14 There

are other ways.

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 → F can be shown to be the projection T ∗ F → F for the cotangent bundle Then N  is thus a holomorphic symplectic manifold) and the projection π : N  →N (and N is a resolution of the singularities on the nilpotent cone N. The map π is semi-small! We know this, for example, from Kaledin’s Theorem 3.1.2. In fact, it was known much earlier, by the work of many. We can partition N according to the Jordan canonical form. This gives rise to a stratification of N and of π. Every stratum of this stratification turns out to be relevant for the semi-small map π. It is amusing to realize that the intersection form associated with the deepest stratum (vertex) gives rise to ± the Euler number of F and that the one associated with the codimension two stratum yields (−1)times the Cartan matrix for sln (C). The fibers of the map π are called Springer fibers. Springer proved that all the irreducible representations of the Weyl group occur as direct summands of the action of the Weyl group on the homology of the Springer fibers. This beautiful result tells us that indeed one can realize geometrically such representations. Note that the Weyl group does not act algebraically on the Springer fibers. In what follows we aim at explaining how the Weyl group acts on the perverse sheaf Rπ∗ QN  . In turn, by taking stalks, this explains why the Weyl group acts on the homology of the Springer fibers.  → N, we consider p : sl Instead of sticking with π : N n (C) → sln (C) defined by the same kind of incidence relation. The difference is that if we take the Zariski open set U given by diagonalizable matrices with n distinct eigenvalues, then p is a topological Galois cover with group the Weyl group (permutation of eigenvalues). We have a Weyl group action! Unfortunately N ∩ U = ∅! On the other hand, p is . . . small! So Rp∗ Q = IC(L), where L is the local system on U associated with the Galois cover with group the Weyl group. Then W acts on L. Hence it acts on IC(L) by functoriality of the intermediate extension construction. Since the map p is proper, and it restricts to π over N, we see that the restriction of Rp∗ Q = IC(L) to N is Rπ∗ Q which thus finds itself endowed, almost by the trick of a magician, with the desired W action! 3.6. Exercises for Lecture 3 Exercise 3.6.1. (Semi-smallness and fibers) Show that, for any maps f : X → Y, we always have dim X ×Y X  dim X. Use Chevalley’s result on the upper semicontinuity of the dimensions of the fibers of maps of algebraic varieties to produce  a finite disjoint union decomposition Y = k0 Sk into locally closed subvari−1 eties with dim f (y) = k for every y ∈ Sk . Show that f is semi-small iff we have dim Sk + 2k  dim X15 for every k  0. Observe that f semi-small implies that f is generically finite, i.e. that S0 is open and dense. Observe that f−1 (S0 ) ×S0 f−1 (S0 ) 15 Think

of it as a vary special upper bound on the dimension of the “stratum” where the fibers are k-dimensional.

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Perverse sheaves and the topology of algebraic varieties

has dimension dim X. Give examples of semi-small maps where X ×Y X has at least two irreducible components of dimension dim X. Exercise 3.6.2. (Semi-small maps and perverse sheaves) A proper map f : X → Y with X nonsingular is semi-small if and only if f∗ QX [dim X] is perverse on Y. Exercise 3.6.3. (Relevant locally constant sheaves) Let Sa ∈ Arel be relevant. Show that Rdim X−dim Sa is locally constant on Sa with stalks H2 dim f

−1 (s)

(f−1 (s)).

The monodromy of this locally constant sheaf, denoted by La , factors through the finite group of symmetries of the set of irreducible components of maximal dimension 12 (dim X − dim Sa ) of a typical fiber f−1 (s). (Note that, a priori, the monodromy could send the fundamental class of such a component to minus itself, thus contradicting the claim just made; that this is not the case follows, for example, from a theorem of Grothendieck’s in EGA IV, 15.6.4. See the nice general discussion in B.C. Ngo’s preprint [44, §7.1.1,], a discussion which we could not locate in the published version [45]). In particular, La is semi-simple. Note that if we switch from Q-coefficients to Q-coefficients simple objects may split further. Do they stay semi-simple? Exercise 3.6.4. (Formalism of correspondences in products) Unwind the isomorphisms (3.4.2) to deduce that under them, a class Γ ∈ Hγ (X × X) defines a linear map Γ∗ : H∗ (X) → H∗+γ−2 dim X (X) given by a → PD(pr2 ∗ (pr∗1 a ∩ Γ )). Conversely, show that any graded linear map H∗ (X) → H∗+γ−2 dim X (X) is given by a unique such Γ ∈ Hγ (X × X). Exercise 3.6.5. (Sheaf theoretic definition of Borel-Moore homology) Let X be a variety. Recall (one) definition of Borel-Moore homology: embed X as a closed subvariety of smooth variety Y, then set (X) := H2dim Y−i (Y, Y − X), HBM i where the right-hand-side is relative cohomology. By interpreting relative cohomology sheaf theoretically, give a sheaf theoretic definition of Bore-Moore homology. (Hint: consider the distinguished triangle i∗ i! → id → Rj∗ j∗ →, if you apply this to the constant sheaf and take cohomology (or equivalently push to a point), what long exact sequence do you get?) Exercise 3.6.6. (Borel-Moore homology and Ext/convolution algebras) Fix a proper morphism of varieties π : X → Y, with X smooth. Form a cartesian square Z p2

 X

p1

/X

π

 /Y

π

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For sheaves A, B on a variety Z, let Exti (A, B) = HomDb (Z) (A, B[i]). I.e., Ext∗ denotes (shifted) Hom in the derived category. Show the following: (1) (2) (3) (4)

Ext∗ (Rπ∗ Q, Rπ∗ Q) = Ext∗ (Q, π! Rπ∗ Q). (Hint: adjunction property). Ext∗ (Q, π! Rπ∗ Q) = Ext∗ (Q, Rp2∗ p1! Q). (Hint: proper base change). Ext∗ (Q, Rp2∗ p1! Q) = Ext∗ (Q, p1! Q) (Hint: push-forward and hom). Ext∗ (Q, p1! Q) = HBM 2dim X−∗ (Z). (Hint: use the sheaf-theoretic definition of Borel-Moore homology; the dimensional shift suggests the use of some kind of duality).

Exercise 3.6.7. (Geometric basis for HBM 2 dim X (X ×Y X) when f is semi-small) Show that if f is semi-small, then the rational vector space HBM 2 dim X (X ×Y X) has a basis formed by the fundamental classes of the irreducible components of X ×Y X of maximal dimension dim X. Describe these irreducible components in terms of monodromy over the relevant strata. Exercise 3.6.8. (Hodge-theoretic decomposition theorem for f semi-small) Show that if f : X → Y is semi-small with X projective nonsingular, then the decomposition H∗ (X, Q) = ⊕a∈Arel IH∗−codim(Sa ) (Sa , La ) is one by pure Hodge structures (i.e. compatible with the Hodge (p, q)-decomposition of H∗ (X, C). (Hint: the projectors onto the direct sums are given by algebraic cycles.)

. Lecture 4: Symmetries: VD, RHL, IC splits off Summary of Lecture 4. Discussion of the two main symmetries in the decomposition theorem for projective maps: Verdier duality and the relative hard Lefschetz theorem. Hard Lefschetz in intersection cohomology and Stanley’s theorem for rational simplicial polytopes. A proof that the intersection complex of the image is always a direct summand. Pure Hodge structures on the intersection cohomology of a projective surface. Remark 4.0.1. We are going to discuss two symmetries for projective maps: Verdier duality and the relative Hard Lefschetz theorem. Both these statements have to do with direct image perverse sheaves. In fact, if the target is projective, then we can take the shadow of these two symmetries in cohomology and notice that there are two additional symmetries: Verdier duality and Hard Lefschetz theorem on the individual summands IH∗ (S, L). Exercise 4.7.7 asks you to make an explicit list in a low-dimensional case. 4.1. Verdier duality and the decomposition theorem Verdier duality and the decomposition theorem. Recall the statement (1.6.3) of the decomposition theorem  ∼ ICS (L)[−q]. Rf∗ ICX (M) = q0,EVq

Switching to the perverse intersection complex. If (S, L) is an enriched variety, then ICS (L) is not a perverse sheaf. Recall from (1.6.1) that the perverse object is

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Perverse sheaves and the topology of algebraic varieties

ICS (L) := ICS (L)[dim S]. In order to emphasize better certain symmetric aspects of the decomposition theorem, we switch to the perverse intersection complex. This entails a a minor headache when re-writing (1.6.3), which becomes (verify it as an exercise)  ∼ ICS (L)[−b], (4.1.1) Rf∗ ICX (M) = b∈Z EVb

where (S, L) ∈ EVb if and only if (S, L) ∈ EVb+dim X−dim S . The perverse cohomology sheaves of the derived direct image. Note that, now, every b-th direct summand above is a perverse sheaf, so that, in view of Exercise 4.7.8, we have that: (4.1.2)

Hb (Rf∗ ICX (M)) = ⊕EVb ICS (L).

p

The case when M is self-dual (and semi-simple). Let M be self-dual, e.g. a constant sheaf, a polarizable variation of pure Hodge structures, or even the direct sum of any M with its dual; self-dual local systems appear frequently in complex algebraic geometry. Then so is ICX (M) and, by the duality exchange property for proper maps, so is Rf∗ ICX (M). In view of the duality relation ∼ (2.5.2) between perverse cohomology sheaves, we see that pHb (Rf∗ ICX (M))) = pH−b (Rf IC (M)))∨ . By combining with (4.1.2), we get ∗ X       ∼ (4.1.3) Rf∗ ICX (M) = ICS (L)[−b] ⊕ ICS (L) ⊕ ICS (L∨ )[b] . b g − 2 such that the deq ∼ composition theorem over Aell is of the form Rhell ∗ Q = ⊕q0 ICAell (R )[−q] (in [13], we reach this conclusion directly and complement it by showing that these intersection complexes are in fact sheaves on Aell ). A generic line will avoid the small closed complement (at least if g  3; g = 2 can be dealt with separately), where Rq are the locally constant direct image sheaves over the regular part. Pick a generic line Λ and observe that the decomposition theorem for the Hitchin map restricted over the line reads:  ∼ ICΛ (Rq )[−q]; RhΛ ∗ Q= q0

this is because the restriction of an (un-shifted) intersection complex IC to a general linear section is an (un-shifted) intersection complex.

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Let j : Λreg → Λ be the open immersion of the set of regular values of hΛ . Since Λ is a nonsingular curve, we know that  ∼ j∗ Rq [−q]; RhΛ ∗ Q = q0

this is because intersection complexes on nonsingular curves are obtained via the ordinary sheaf-theoretic push-forward (Fact 1.7.19). Note that our perverse sheaves are now just sheaves (up to shift). It follows that the perverse spectral sequence for hΛ is just the ordinary Leray spectral sequence and the same holds for map hΛreg over the set of regular values of hΛ . By the functoriality of the Leray spectral sequence and by Artin vanishing on the affine curves Λ and Λreg (H>1 = 0!), we have a commutative diagram 0 (5.4.4) 0

/ H1 (Λ, j∗ R3 )

/ H4 (MΛ )

/ H0 (A, j∗ R4 )



 / H4 (MΛ ) reg

 / H0 (R4 )

/ H1 (Λreg , R3 )

/0

=

/ 0,

of short exact sequences (the edge sequences for the Leray spectral sequences for the maps h) where the first vertical map is injective (edge sequence for the Leray spectral sequence for the map j), and the third is an isomorphism (definition of direct image sheaf). A simple diagram chase, tells us that the vertical restriction map in the middle of (5.4.4) is also injective. On the other hand, the class β|MΛ → β|MΛreg = 0 by what seen earlier (β restricts to zero over Areg , hence over Λreg ). By the injectivity statement above, we see that β vanishes over the generic line and we deduce that β ∈ PD,2 H4 (MD , Q), as predicted by P = W. Question 5.4.5. We can formulate P = W for every complex reductive group. Does it hold, at least for GLn ? There are indications that this should hold for GLn , n small. 5.5. Let us conclude with a motivic question Let f : X → Y be a projective map of projective varieties with X nonsingular. By the decomposition theorem, there is an isomorphism (5.5.1)

∼ ⊕q,EV IH∗−q (S, L). φ : H∗ (X, Q) = q

This implies that for each (S, L) in EVq we obtain a projector (map that squares to itself) on H∗ (X, Q) with image φ(IH∗−q (S, L)). We view this projector as a cohomology class πφ := H2 dim X (X × X, Q). It is possible to endow each term on the r.h.s. of (5.5.1) with a natural pure Hodge structure and then to choose an isomorphism φ (5.5.1) that is an isomorphism of pure Hodge structures. This implies that πφ is rational and that is has (p, q)-type (dim X, dim X), i.e. it is a Hodge class.

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According to the Hodge conjecture, πφ should be algebraic (cohomology class of an algebraic cycle in X × X). Question 5.5.2. (Motivic decomposition theorem) Can we chose φ so that the resulting projectors πφ are given by algebraic cycles? The answer is positive for semi-small maps (Exercise 3.6.8). We have no idea if/why this should be true. We can prove something much weaker: the projectors are absolute Hodge (in the sense of Deligne), even motivated (in the sense of André); see [23]. If the answer were true, then applying it to the blowing up of the projective cone over an embedded projective manifold would yield the Grothendieck standard conjecture of Lefschetz type (the inverse to the Hard Lefschetz isomorphisms are induced by algebraic cycles in the product); see the introduction to [8]. 5.6. Exercises for Lecture 5 Exercise 5.6.1. (Restricting perverse sheaves to general linear sections) Let Y be quasi projective and P ∈ P(Y). Show that if i : Yk → Y is a codimension k general linear section of Y relative to any fixed embedding in Y → PN , then P|Yk [−k] ∈ P(Yk ). To do so first verify directly that P|Yk [−k] satisfies the conditions of support (use the Bertini theorem to cut down the supports of cohomology sheaves). It is not so trivial to verify the conditions of co-support, i.e. the conditions of support for the dual of P|Yk [−k]. Here, we simply say that we can choose Yk general, depending on P, so that i! = i∗ [−2k] and that the desired conditions follow formally from this and from the duality exchange property: (i∗ (P[−k]))∨ = i! P∨ [k] = i∗ P∨ [−k]. (See [16], for example). Exercise 5.6.2. (Restriction to general linear sections and maps in cohomology) Let Y be quasi projective, let P ∈ D(Y) and let Yk be the complete intersection of k general linear sections of Y relative to any embedding of Y in some projective space. Use the Lefschetz hyperplane theorem to show that the restriction maps H∗ (Y, P) → H∗ (Yk , P|Yk ) are injective for every ∗  −k. Assume in addition that Y is affine. Use the Artin vanishing theorem for perverse sheaves and Exercise 5.6.1 to show that the same restriction maps are zero for ∗ > −k. Exercise 5.6.3. (Geometric description of perverse Leray) Write out explicitly the conclusion of Theorem 5.2.5 in the case when f is the blow-up at the vertex of the affine cone over P1 × P1 and verify it. Exercise 5.6.4. (Jouanolou’s trick) Let Y be quasi projective. There is a map p : Y → Y with Y affine and that is a Zariski locally trivial bundle with fiber affine spaces Am . There are various different ways to do this; following is a series of hints to reach this goal. Choose a projective completion Y ⊆ Y so that the embedding is affine (e.g. with boundary a divisor). Argue that we may assume that Y is projective. Pick a suitably positive rank dim Y + 1 vector bundle E on

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Y, where suitably positive here means that the associated line bundle OP(E) (1) is very ample and has a section s that is not identically zero on any projective fiber. Show that taking P1 × P1 \ Δ → P1 yields a special case of the construction above. Show that Y := P(E) \ (s = 0) does the job. The usefulness of this trick in our situation is that if we start with f : X → Y, with Y quasi projective, we can base change to g : X → Y so that now the target is affine and the properties of p (smooth map with “contractible” fibers) allow us to prove the assertions on f by first proving them for g and then “descending” them to f. Exercise 5.6.5. (Leray and perverse Leray) In this exercise use the following: if j : So → S is an open embedding of nonsingular curves and L is a locally constant sheaf on So , then ICS (L) = R0 j∗ L. Let p : P1 × P1 → P1 be a projection and let b : X → P1 × P1 be the blowing up at a point. Let f := p ◦ b. Determine and compare the perverse Leray and the Leray filtrations for f on H∗ (X, Q). (Renumber the perverse Leray one so that 1 ∈ P0 \ P−1 ; this way they both “start” at the same “time”.) Do the same thing, but for a Lefschetz pencil of plane curves. Note how the graded spaces for the two filtrations differ in the “middle”. Exercise 5.6.6. (H∗ (MD ) is pure). There is a natural C∗ -action on MD obtained by multiplying the Higgs field φ by a scalar. Show that the Hitchin map is equivariant for this action. Use the C∗ -action and the properness of the Hitchin map to show that the closed embedding of the fiber h−1 (0) → MD induces an isomorphism in cohomology. Use the weight inequalities listed at the end of our quick review of mixed Hodge theory in §1.7, to deduce the Hj (MD ) is pure of weight j for every j. Exercise 5.6.7. (Normalizing the perverse filtration) Assume that C = ⊕b Pb [−b] ∈ D(Y),

with Pb ∈ P(Y).

Show that Pb H∗ (Y, C[m])) = Pb+m H∗+m (Y, C) (in fact, this is true in general, i.e. without assuming that C splits). Deduce that if H∗ (Y, C) = 0, then there ∗ P ∗ is a unique m ∈ Z such that GrP b0 SL2 ). Then Km /Km = SL2 . Now let iα : SL2 → G be the canonical homomorphism corresponding to α. Let m = (μ, α) − 1 and consider the orbit Cμ,α := Liα (Km )tμ .

(2.1.6) (1)

Note that Km ⊂ L+ G ∩ tμ L+ Gt−μ so that Cμ,α is a homogenous space under (1) ∼ P1 with (L+ G ∩ Liα (Km ))tμ = ∼ A1 ⊂ P1 . In Km /Km = SL2 and that Cμ,α =

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addition, P1 \ A1 = iα (σm )tμ . From the famous identity       1 0 1 −t−1 t−1 0 −1 1 −t−1 = 0 1 t 1 0 1 0 1 0

 0 t

,

we see that iα (σm )tμ = (tμ−α mod L+ G). Thus, Cμ,α is the desired curve.



Remark 2.1.7. Assume, for simplicity, that Gder is simply connected and char k = 0.  ⊂ Gr whose R-points clasThen one can define a closed subscheme of Grμ sify those (E, β) such that for every irreducible highest weight representation Vχ , β(Vχ,E ) ⊂ t−(χ,μ) (Vχ ⊗ R[[t]]). From the proof of Proposition 2.1.4, we know that  . But it is an open question to whether Grμ is the reduced subscheme of Grμ  Grμ = Grμ . This is closely related to [42, Conjecture 2.14, Conjecture 2.20]. Example 2.1.8. Let G = GLn and μ = (r, 0, . . . , 0) (under the standard identification ∼ Zn ). Then using the lattice description of GrGL as in §1.1, of X• (T ) = n Grμ (R) = {Λ ⊂ Λ0 := R[[t]]n | rk Λ0 /Λ = r} . Now assume that r = n. Let Nn denote the variety of n × n nilpotent matrices. There is a natural map (2.1.9)

Nn → Grμ ,

A → Λ = (t − A)Λ0 .

This is well-defined since det(t − A) = tn . We will show in Lemma 2.3.13 that this is an open embedding so, in particular, Grμ gives a compactification of Nn . At the level of topological spaces, this was first observed by Lusztig (cf. [50]). Note that giving a dominant coweight λ  μ = (n, 0, . . . , 0) is equivalent to giving a partition of n. So each λ  μ gives a nilpotent orbit Oλ in Nn . We leave it as an exercise to show that (2.1.9) maps Oλ to Grλ . We continue the general theory. There is a map (2.1.10)

p : X• (T ) → Z/2, μ → (−1)(2ρ,μ)

which factors through the composition X• (T ) → π1 (G) → Z/2 and therefore induces a map p : π0 (GrG ) → Z/2 by Theorem 1.3.11. Corollary 2.1.11. The Schubert cell Grμ is in the even (resp. odd) components, i.e. p(Grμ ) = 1 (resp. p(Grμ ) = −1) if and only if dim Grμ is even (resp. odd). In the sequel, let (LG)μ (resp. (LG)μ ) denote the preimage of Grμ (resp. Grμ ) in LG. Note that (LG)μ and (LG)μ are schemes. For a coweight μ, let Pμ denote the parabolic subgroup of G corresponding to μ, i.e. the group generated by the root subgroups Uα of G for those roots α satisfying α, μ  0. Let us denote the evaluation map by g → g¯ := ev(g) (see Example 1.3.4). Then there is a natural projection ∼ L+ G/(L+ G ∩ tμ L+ Gt−μ ) → G/Pμ by (gtμ mod L+ G) → (g¯ mod Pμ ) . pμ : Grμ =

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The fibers are isomorphic to affine spaces. On the other hand, there is also a closed embedding ∼ G · tμ ⊂ Grμ iμ : G/Pμ = by regarding G as the subgroup of L+ G of “constant loops”. Note that the action of Aut+ (D) on GrG preserves Grμ 3 and the fixed point subscheme under the μ rotation torus Grot m is G · t . Example 2.1.12. Let G = GLn , let Λ0 = k[[t]]n and let μ = (m1 , . . . , mn ) ∈ Zn be a dominant coweight (so mi  mi+1 ). Then the map pμ : Grμ → G/Pμ sends a ∼ kn . lattice Λ ∈ Grμ to the following decreasing filtration V = Λ0 /tΛ0 = Fili V = t−mi Λ ∩ Λ0 /t−mi Λ ∩ tΛ0 . Conversely, given a decreasing filtration Fil• on V, the map iμ sends it to the lattice  ∼ Fn . Λ= Fili V ⊗ tmi O ⊂ V ⊗k F = We give more explicit descriptions of Grμ for small μ. Recall that a dominant coweight μ is called minuscule if μ = 0 and for every positive root α, (α, μ)  1. Note that minuscule coweights are minimal elements in X• (T )+ under the partial order . Therefore, ∼ G/Pμ . Lemma 2.1.13. If μ is minuscule, then Grμ = Grμ = So minuscule Schubert varieties in GrG are exactly those partial flag varieties classifying parabolic subgroups of G corresponding to minuscule coweights. Examples include the usual Grassmannians Gr(r, n), smooth quadrics in projective spaces, etc. Recall that a dominant coweight μ is called quasi-minuscule if μ = 0, μ is not minuscule, and for every positive root α, (α, μ)  2. For simplicity, we assume that Gder is simple. Then μ is the short dominant coroot, denoted by θ, and is the minimal element in X• (T )+ − {0} under the partial order . The root corresponding to θ is the highest root θ∨ . Let Lθ = G ×Pθ kθ∨ be the very ample line bundle on G/Pθ , where Pθ acts on the 1-dimension space kθ∨ by the character θ∨ . Note that 0 ∈ X• (T )+ is the unique element that is strictly less than θ. Therefore, Grθ = Grθ  Gr0 . Lemma 2.1.14. Assume (for simplicity) that Gder is simple. If μ = θ is quasi-minuscule, then via the projection pθ , Grθ is isomorphic to the total space of the line bundle Lθ , and Grθ is isomorphic to the projective cone over the projective embedding of G/Pθ by Lθ .  θ = P(Lθ ⊕ O) gives a resolution of singularities . Blowing-up at the origin Gr Proof. See [58, §7].



 θ is a kind of Demazure resolution of Grθ . See [84, Remark 2.1.15. In fact, Gr Lemma 2.12]. the action of Aut(D) on GrG does not preserve Schubert varieties as gtμ ∈ Grμ for g ∈ Aut(D) sending t → a0 + t, where a0 ∈ R is nilpotent.

3 However,

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Example 2.1.16. If G = SL2 , then θ = α is the positive coroot. In this case Grθ is isomorphic to the projective cone over the quadratic curve {(x, y, z) | x2 = yz} in P2 . Given μ1 , μ2 ∈ X+ • of G, one can define the twisted product of Grμ1 and + Grμ2 using the L G-torsor (LG)μ1 → Grμ1 . Alternatively, one can define it as ˜ μ2 = {(E1 , E2 , β1 , β2 ) ∈ Gr×Gr ˜ | Inv(β1 )  μ1 , Inv(β2 )  μ2 } , Grμ1 ×Gr ˜ which is closed in Gr×Gr and therefore is representable. Similarly, given a sequence μ• = (μ1 , . . . , μn ) of dominant coweights of G, one can define ˜ · · · ×Gr ˜ μn ⊂ Grμ• := Grμ1 × ˜ · · · ×Gr ˜ μn ⊂ Gr× ˜ · · · ×Gr. ˜ Grμ• := Grμ1 ×  Let |μ• | = μi , then the convolution map (1.2.15) induces (2.1.17)

m : Grμ• → Gr|μ• | ,

(E• , β• ) → (En , β1 · · · βn ).

This map is analogous to Demazure resolutions (except that Grμ• is not smooth in general). The geometry of these maps is very rich, as we shall see in §5. Here we give one example. Example 2.1.18. We use notations from Example 2.1.8. Now, let ω1 = (1, 0, . . . , 0). We have the convolution map m : Gr(ω1 ,...,ω1 ) → Grμ . On the other hand, ˜ r → Nr (cf. [80, §1]). These maps fit there is the classical Springer resolution p : N into the following commutative diagram

(2.1.19)

Gr(ω1 ,...,ω1 ) ←−−−− Gr (ω1 ,...,ω1 ) −−−−→ ⏐ ⏐ ⏐ ⏐ m  Grμ

π

←−−−−

Gr μ

˜r N ⏐ ⏐p 

φ

−−−−→ Nr ,

where Gr μ (R) classifies triples (E, β, ) in which (E, β) ∈ Grμ (R) and in which ∼ Rr is an isomorphism of R-modules. Note that both squares in the  : Λ0 /Λ = diagram are Cartesian. Indeed, any point of Gr(ω1 ,...,ω1 ) over Λ ∈ Grμ gives us a chain Λ = Λr ⊂ Λr−1 ⊂ · · · ⊂ Λ0 = k[[t]]n . Via the framing , such a chain gives a full flag of kr and vice versa. Now we assume r = n. Then the formula (2.1.9) in fact gives a section Nn → Gr μ of φ (since (t − A)Λ0 is canonically trivial). Therefore, we obtain the following Cartesian diagram

(2.1.20)

˜ n −−−−→ Gr(ω ,...,ω ) N 1 ⏐ ⏐1 ⏐ ⏐ p m Nn −−−−→

Grμ .

In other words, the convolution map m in this case extends the Springer resolution for GLn .

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We continue the general discussion. The following fundamental theorem summarises the basic facts of the singularities of Schubert varieties. Theorem 2.1.21. Let p denote the characteristic exponent of k. If p  |π1 (Gder )|, then Grμ is normal, Cohen-Macaulay, Gorenstein, and has rational singularities. Proof. This is a difficult theorem. Except the Gorenstein property, all statements were proved in [24, Theorem 8]; see also [61, Theorem 0.3]. The key step is to prove that Grμ is normal. That Grμ is Gorenstein follows from [11, Equation (241)]. Strictly speaking, the argument there only works when the characteristic of k is zero (or is very good for G). We refer to [82, Theorem 6.11] for more details and the extension to  the case p  |π1 (Gder )|. See also Remark 2.3.11 for related discussions. Remark 2.1.22. We very briefly discuss the parallel story when G is an Iwahori group scheme of G. See [24, 61] for details. Write I = G(O). In this case, the Cartan decomposition is of the form  I\G(F)/I  W,   X• (T )  W is called the Iwahori Weyl group. This is a (quasi-)Coxeter where W  → Z. For group equipped with a Bruhat order  and a length function  : W +  let Sw denote the corresponding L G-orbits on GrG = FG and let w ∈ W, Sw denote its closure. Then Sw is isomorphic to an affine space of dimension  (w), and Sw = w  w Sw . All Sw ’s are normal, Cohen-Macaulay, and with rational singularities. But they are not Gorenstein in general. 2.2. Digression: Some sub-ind-schemes in Gr We make a digression to explain a construction of some important subvarieties of Gr. This subsection is not used in the rest of the notes. Let ϕ : Gr → Gr be a morphism defined over k. For γ ∈ LG and μ given, we define a closed sub ind-scheme of Gr as X(μ, γϕ) = {x ∈ Gr | Inv(x, γϕ(x))  μ} . Explicitly, write x = gL+ G ∈ Gr. Then

X(μ, γϕ) = gL+ G | g−1 γϕ(g) ∈ (LG)μ /L+ G. Here are a few equivalent definitions. (1) X(γϕ, μ) is defined by the following Cartesian diagram ˜ μ X(μ, γϕ) −−−−→ Gr×Gr ⏐ ⏐ ⏐ ⏐   1×γϕ

−−−−→ Gr × Gr. Gr (2) Assume that ϕ is induced by a morphism ϕ : LG → LG that preserves L+ G. We consider the conjugate action of LG twisted by ϕ. Explicitly,

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Int(g) ·ϕ g  = g−1 g  ϕ(g). Consider the morphism of stacks π : [(LG)μ /ϕ L+ G] → [LG/ϕ LG]. Then X(μ, γϕ) = π−1 (γ). Remark 2.2.1. In many cases, the quotient [LG/ϕ LG], although not well-behaved, should be thought as the moduli space of certain important algebraic structures. Here are some concrete examples of the above construction. (i) Let ϕ = id be the identity map. Then X(γ, μ) is a group version affine Springer fiber (cf. [80, §2.7.3]). In this case, [LG/LG] classifies conjugacy classes of G. There is also a Lie algebra analogue of this construction, which gives the (usual) affine Springer fibers (cf. [80, §2.2]). (ii) Assume that k = Fq and let ϕ = σ be the q-Frobenius of Gr. Then X(μ, γϕ) is usually denoted by X(μ, b) and is called the affine DeligneLusztig variety (first introduced in [66]). In this case [LG/σ LG] classifies ¯ σ-conjugacy classes of G(k((t))), which are closely related to Drinfeld Shtukas. (iii) Now assume that k = Fp . Let ϕ : Gr → Gr be the morphism induced  p ip  ai t . Write γϕ by by the absolute Frobenius k[[t]] → k[[t]], ai ti → Φ. Then X(Φ, μ) is called the Kisin variety, and [LG/σ LG] classifies Galois representations Gal(F/F) → G(Fp ) (see [62]). (iv) Assume that k = C and let ϕ be induced by the loop rotation t → qt, ϕ for some q ∈ Grot m . Then [LG/ LG] is closely related to G-bundles on the × Z elliptic curve Eq = C /q (Looijenga, Baranovsky and Ginzburg [2]). ++ (D). A Lie algebra (v) Instead of taking ϕ ∈ Grot m , one can let ϕ ∈ Aut analogue of this construction was studied in [27]. In each case, some basic questions to ask are (1) When is the variety finite dimensional? (2) If it is finite dimensional, is there a dimension formula? Are the irreducible components equidimensional? (3) How do we parameterise the connected components? (4) How do we parameterise the irreducible components? These questions are better understood for Case (i) and (ii) (e.g. see [34, 80] and the references cited there for a summary of known results), but remain mostly untouched in other cases. Remark 2.2.2. Thanks to §1.7, affine Springer fibers and affine Deligne-Lusztig varieties now are also defined for p-adic groups. 2.3. Opposite Schubert “varieties” and transversal slices Opposite Schubert “varieties” are very different from the Schubert varieties Grμ introduced in §2.1. In fact, they are not really varieties and are infinite dimensional (but finite codimensional).

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First, let L− G be the presheaf of groups defined as L− G(R) = G(R[t−1 ]). In §2.5, we shall see that LG is closely related to a Kac-Moody group. From this point of view, L− G then is a “parabolic opposite” to L+ G. There is another point ∗ of view. Namely, L− G is a special case of the group GX which will be introduced in (4.1.3) (corresponding to the case X = P1 and x = 0). In particular, by Lemma 4.1.4 below, the group L− G is represented by an ind-scheme. We fix T ⊂ B ⊂ G as in §2.1. Then there is the Birkhoff decomposition of G(F) G(k[t−1 ])tμ G(O), (2.3.1) G(F) = μ∈X• (T )+

which induces (2.3.2)

∼ X• (T )+ . L− G(k)\G(F)/G(O) =

Similar to the Cartan decomposition (2.1.1), (2.3.2) is independent of any choice. We refer to [24, Lemma 4] for a proof of this decomposition for general G. We define opposite Schubert “cells” as L− G-orbits on Gr. By the Birkhoff decomposition (2.3.2), they are of the form Grμ = (L− G · tμ )red ⊂ Gr, and are parameterised by X• (T )+ . Let Grλ , Grμ = λμ

called the opposite Schubert variety. The terminology is justified by the following proposition. Proposition 2.3.3. (1) Grμ is a locally closed sub-ind-scheme of Gr. ∼ G/Pμ , (2) Grμ ∩ Grλ = ∅ if and only if μ  λ. In addition, Grμ ∩ Grμ = G · tμ = rot which is fixed by the rotation torus Gm . (3) Grμ is Zariski closed, and contains Grμ as an open dense subset. (4) The codimension of Grμ is (2ρ, μ) − dim G/Pμ . Proof. (1) The stabiliser of tμ in L− G is L− G ∩ tμ L+ Gt−μ , which is a finite dimension subgroup. By writing L− G = limKi such that Ki ⊃ L− G ∩ tμ L+ Gt−μ , we see −→ μ that each Ki tμ is locally closed in Gr and Grμ = lim −→(Ki t )red is an ind-scheme. We also note that (2.3.4)

dim(L− G ∩ tμ L+ Gt−μ ) = (2ρ, μ) + dim Pμ ,

which follows by calculating the dimension of its Lie algebra as in the argument for Proposition 2.1.5. (2) Recall that the rotation torus acts on Gr and the fixed point set of this action  is λ∈X• (T )+ G · tλ . Since Gr is ind-projective, for every x ∈ Gr, the action map

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1 ax : Grot m → Gr by r → r · x extends to a map P → Gr, still denoted by ax . Then it is easy to see that   Grμ = x ∈ Gr | ax (0) ∈ G · tμ , Grλ = x ∈ Gr | ax (∞) ∈ G · tλ .

Now let x ∈ Grμ ∩ Grλ . Since Grμ is projective, ax (∞) ∈ Grμ . Therefore, λ  μ. Conversely, assume that λ  μ. Again, by [65, Lemma 2.3], we can write μ = μ0  μ1  · · ·  μr+1 = λ such that each μi is dominant and αi = μi − μi+1 is a positive coroot. Let mi = (μi , αi ) − 1. Recall that for every positive root α, we denote by iα : SL2 → G the induced map. We consider the map f : A1 → Gr sending y to       1 ytmr 1 ytm1 1 ytm0 μ · · · iα 1 iα 0 t . iα r 1 1 1 Its extension to P1 → Gr is still denoted by f. Note that this curve is contained in Liαr (Kmr ) · · · Liα0 (Km0 )tμ , where Km ⊂ LSL2 was defined in the proof of Proposition 2.1.5. It follows from arguments there that f(∞) = tλ . Note that μ f(P1 ) is stable under the Grot m -action. Therefore, for y = {0, ∞}, af(y) (0) = t , and λ λ af(y) (∞) = t . It follows that f(y) ∈ Grμ ∩ Gr . (3) We first study Gr0 . Note that there is an evaluation map L− G → G,

g → g mod t−1 .

We define L 1. Take the 2 d second largest stratum Bund−1 G ×C . By the factorization, UG is isomorphic to a 1 product of a smooth variety and UG in a neighborhood of a point in the stratum. Fu’s argument is local (the minimal nilpotent orbit has an isolated singular point), hence Ud G does not have a symplectic resolution except when G = SL(r). 2.5. Group action We have an action of a group G on Ud G by change of framing. induced from the GL(2)-action on C2 . Let We also have an action of GL(2) on Ud G T be a maximal torus of G. In these notes, we only consider the action of the subgroup T × C× × C× in G × GL(2). Let us introduce the following notation: T = T × C× × C× . Our main player is the equivariant intersection cohomology group IH∗T (Ud G ). It is a module over (2.5.1)

∼ C[Lie T] = ∼ C[ε1 , ε2 , a], H∗T (pt) =

where a  (resp. ε1 and ε2 ) are coordinates (called equivariant variables) on Lie T (resp. Lie(C× × C× )). We also use the notation a, ε1 , ε2 ], AT = C[Lie T] = C[

A = C[ε1 , ε2 ].

6 The

author does not know who first noticed this fact, and even where a proof is written. In [2] it was shown that any Gc 1-instanton is reduced to an SU(2) 1-instanton, and the instanton moduli space (but not framed one) for SU(2) is determined. But this statement itself is not stated.

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Lectures on perverse sheaves on instanton moduli spaces

Their quotient fields are denoted by FT = C(Lie T),

F = C(ε1 , ε2 ).

3. Heisenberg algebra action on the Gieseker partial compactification This lecture is an introduction to the actual content of subsequent lectures. We consider instanton moduli spaces when the gauge group G is SL(r). We will explain results about (intersection) cohomology groups of instanton moduli spaces known before the AGT correspondence was found. Then it will be clear what was lacking, and readers will be motivated to learn more recent works. 3.1. Gieseker partial compactification When the gauge group G is SL(r), we d denote the corresponding Uhlenbeck partial compactification Ud G by Ur .  d of Ud , called the Gieseker partial For SL(r), we can consider a modification U r r compactification. It is a moduli space of framed torsion free sheaves (E, ϕ) on P2 , where the framing ϕ is a trivialization of the restriction of E to the line at infinity  d is a smooth (holomorphic) symplectic manifold. It is also ∞ . It is known that U r  d → Ud , which is a resolution of known that there is a projective morphism π : U r r singularities.  d is When r = 1, the group SL(1) is trivial. But the Giesker space is nontrivial: U 1 the Hilbert scheme (C2 )[d] of d points on the plane C2 , and the Uhlenbeck partial th symmetric product Sd (C 2 ) of C 2 . The former compactification7 Ud 1 is the d parametrizes ideals I in the polynomial ring C[x, y] of two variables with colength d, i.e., dim C[x, y]/I = d. The latter is the quotient of the Cartesian product (C2 )d by the symmetric group Sd of d letters. It parametrizes d unordered points in C2 , possibly with multiplicities. We will often use summation notation such as p1 + p2 + · · · + pd or d · p to express a point in Sd (C2 ). Cd

B1

B2 J

I Cr

Figure 3.1.1. Quiver varieties of Jordan type For general r, these spaces can be understood as quiver varieties associated with Jordan quiver. It is not my intention to explain the theory of quiver varieties, but here is the definition in this case: Take two complex vector spaces of dimension d, r respectively. Consider linear maps B1 , B2 , I, J as in Figure 3.1.1. We impose the equation def.

μ(B1 , B2 , I, J) = [B1 , B2 ] + IJ = 0 7 Since

Bund GL(1) = ∅ unless d = 0, this is a confusing name.

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Then we take two types of quotient of μ−1 (0) by GL(d). The first one corresponds −1 to Ud r , and is the affine algebro-geometric quotient μ (0)//GL(d). It is defined −1 GL(d) , the ring of GL(d)-invariant polynomials on as the spectrum of C[μ (0)] −1 μ (0). Set-theoretically it is the space of closed GL(d)-orbits in μ−1 (0). The  d , and is the geometric invariant theory quotient second quotient corresponds to U r with respect to the polarization given by the determinant of GL(d). It is Proj  n of n0 C[μ−1 (0)]GL(d),det , the ring of GL(d)-semi-invariant polynomials. Settheoretically it is the quotient of the set of stable points in μ−1 (0) by GL(d). Here (B1 , B2 , I, J) is stable if there is no proper subspace T of Cd which is invariant under B1 and B2 and contains the image of I. Theorem 3.1.2. We have isomorphisms ∼ −1 Ud r = μ (0)//GL(d),

⎛

d = ∼ Proj ⎝ U r



⎞ C[μ−1 (0)]GL(d),det ⎠ . n

n0

 d and Ud . Let us briefly recall how these linear maps determine points in U r r Details are given in [43, Ch. 2]. Given (B1 , B2 , I, J), we consider the following complex O⊕d ⊕ ⊕d

O

(−1) − → O⊕d − → O⊕d (1), a

b

⊕ O⊕r where

⎛

z0 B1 − z1

⎞

⎜ ⎟ ⎟ a=⎜ ⎝z0 B2 − z2 ⎠ ,

 b = −(z0 B2 − z2 ) z0 B1 − z1

 z0 I .

z0 J Here [z0 : z1 : z2 ] is the homogeneous coordinate system of P2 such that we have ∞ = {z0 = 0}. The equation μ = 0 guarantees that this is a complex, i.e., ba = 0. One sees easily that a is injective on all but finitely many fibers over [z0 : z1 : z2 ]. The stability condition implies that b is surjective on each fiber and def. that E = Ker b/ Im a is a torsion free sheaf of rank r with c2 = d. Considering the restriction to z0 = 0, one sees that E has a canonical trivialization ϕ there. Thus we obtain a framed sheaf (E, ϕ) on P2 . One also see that a is injective on any fiber if and only if E is a locally free sheaf, i.e., a vector bundle. From this description, we can check the stratification (2.2.1). If (B1 , B2 , I, J) has a closed GL(d)-orbit, it is semisimple, i.e., a ‘submodule’ (in appropriate sense) has a complementary submodule. Thus (B1 , B2 , I, J) decomposes into a direct sum of simple modules, which do not have nontrivial submodules. There is exactly one simple summand with nontrivial I, J, and all others have I = J = 0. The former

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Lectures on perverse sheaves on instanton moduli spaces 

gives a point in Bund G , as one can check that a is injective in this case. The latter is a pair of commuting matrices [B1 , B2 ] = 0, and the simplicity means that the size of matrices is 1. Therefore the simultaneous eigenvalues give a point in C2 . Exercise 3.1.3. (a) We define the factorization morphism πd for G = SL(r) in terms of (B1 , B2 , I, J). Let πd ([B1 , B2 , I, J]) ∈ Sd C be the spectrum of B1 counted with multiplicities. Check that πd satisfies the properties (1),(2) above. (b) Check that F|P1x is trivial if B1 − x is invertible. More generally one can define the projection as the spectrum of a1 B1 + a2 B2 for (a1 , a2 ) ∈ C2 \ {0}, but it is enough to check this case after a rotation by the GL(2)-action.  d . It is a rank d 3.2. Tautological bundle Let V be the tautological bundle over U r vector bundle whose fiber at a framed torsion free sheaf (E, ϕ) is H1 (P2 , E(−∞ )). In the quiver variety description, it is the vector bundle associated with the prin d . For Hilbert schemes of points, parametrizcipal GL(d)-bundle μ−1 (0)stable → U r ing ideals I in C[x, y], the fiber at I is C[x, y]/I. 3.3. Gieseker-Uhlenbeck morphism Recall that a (surjective) projective morphism π : M → X from a nonsingular variety M is semi-small if X has a strati fication X = Xα such that π|π−1 (Xα ) is a topological fibration and, for xα ∈ Xα , dim π−1 (xα )  12 codim Xα .  d → Ud is semi-small with Proposition 3.3.1 ([4], [43, Exercise 5.15]). The map π : U r r respect to the stratification (2.2.2). Moreover, the fiber π−1 (x) is irreducible8 at any point x ∈ Ud r. This semi-smallness result is proved for general symplectic resolutions by Kaledin [28]. Exercise 5.15 in [43] asks for the dimension of the central fiber π−1 (d · 0). Let us explain why the estimate for the central fiber is enough. Take x ∈ Ud r and write   , x =

x . The morphism π assigns it as (F, ϕ, λi xi ), where (E, ϕ) ∈ Bund i j SL(r) ∨∨ ∨∨ d  (E , ϕ, Supp(E /E)) to a framed torsion free sheaf (E, ϕ) ∈ Ur . See [43, Exercise 3.53]. Then π−1 (x) parametrizes quotients of E∨∨ with given multiplicities λi at xi . Then it is clear that π−1 (x) is isomorphic to the product of quotients of O⊕r  with multiplicities λi at 0, i.e., i π−1 (λi · 0). If one knows each π−1 (λi · 0) has di  2 mension rλi − 1, we have dim π−1 (x) = (rλi − 1) = 12 codim Bund SL(r) ×Sλ (C ). −1 Thus it is enough to check that π (d · 0) = rd − 1. 3.4. Equivariant cohomology groups of Giesker partial compactification We  d , Ud as in §2.5. The Gieseker-Uhlenbeck have a group action of G × GL(2) on U r r  d → Ud is equivariant. Let T = T × C× × C× as in §2.5. We will morphism π : U r r study the equivariant cohomology groups of the Gieseker spaces [∗]  d HT (U r ),

[∗]  d HT,c (U r)

8 The (solution of) the exercise only shows there is single irreducible component of π−1 (x ) with α dimension 12 codim Xα . The irreducibility was proved by Baranovsky and Ellingsrud-Lehn.

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with arbitrary and compact support respectively. We use the degree convention, i.e., the degree 0 is 2dr. We denote the equivariant variables by a  = (a1 , . . . , ar ) with a1 + · · · + ar = 0 × × for T , and ε1 , ε2 for C × C . (See (2.5.1).) We have the intersection pairing  [∗]  d [∗]  d ∗  dr HT (U r ) ⊗ HT,c (Ur ) → HT (pt); c ⊗ c → (−1)

d U r

c ∪ c .

This is of degree 0. The sign (−1)dr is introduced to save (−1) in later formulae.  d . Similarly, The factor dr should be understood as the half of the dimension of U r [∗] 1 2 2 since 2 dim C = 1, the intersection form on HC× ×C× (C ) has the sign factor (−1)1 . 3.5. Heisenberg algebra via correspondences Let n > 0. Let us consider a cor d × C2 :  d+n × U respondence in the triple product U r r

def. d+n d   × C2 E1 ⊂ E2 , Supp(E2 /E1 ) = {x} . × U Pn = (E1 , ϕ1 , E2 , ϕ2 , x) ∈ U r r Here the condition Supp(E2 /E1 ) = {x} means the quotient sheaf E2 /E1 is 0 outside x. On the left hand side, the index d is omitted: we understand either Pn is the disjoint union for various d, or d is implicit from the situation. It is known that  d+n × U  d × C2 . Pn is a Lagrangian subvariety in U r r  d+n , q2 : Pn → U  d × C2 , which are proper. We have two projections q1 : Pn → U r r The convolution product gives an operator [∗+deg α]  d+n [∗]  d Δ (α) : HT (U (Ur ); c → q1∗ (q∗2 (c ⊗ α) ∩ [Pn ]) P−n r ) → HT [∗]

for α ∈ HC× ×C× (C2 ). The meaning of the superscript ‘Δ’ will be explained later. We also consider the adjoint operator [∗+deg α]  d [∗]  d+n Δ Δ (α) = (P−n (α))∗ : HT,c (U ) → HT,c (Ur ). Pn r [∗]

A class β ∈ HC× ×C× ,c (C2 ) with compact support gives a pair of operators [∗+deg β]  d+n [∗]  d Δ (β) : HT,c (U (Ur ) , P−n r ) → HT,c

and

[∗+deg β]  d [∗]  d+n Δ Δ (β) = (P−n (β))∗ : HT (U ) → HT (Ur ) . Pn r  [∗]  d Theorem 3.5.1 ([26, 41] for r = 1, [4] for r  2). As operators on d HT (U r ) or  [∗]  d d HT,c (Ur ), we have the Heisenberg commutator relations " ! Δ Δ (α), Pn (β) = rmδm,−n α, β id . (3.5.2) Pm

Here α, β are equivariant cohomology classes on C2 with arbitrary or compact support. When the right hand side is nonzero, m and n have different sign, hence one of α and β has compact support and the other has arbitrary support. Then α, β is well-defined.

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Lectures on perverse sheaves on instanton moduli spaces

Historical Comment 3.5.3. As mentioned in the Introduction, the author [40] found relations between the representation theory of affine Lie algebras and moduli spaces of instantons on C2 /Γ , where the affine Lie algebra is given by Γ by the McKay correspondence. It was motivated by works by Ringel [55] and Lusztig [32], constructing upper-triangular subalgebras of quantum enveloping algebras by representations of quivers. The above theorem can be regarded as the case Γ = {1}, but the Heisenberg algebra is not a Kac-Moody Lie algebra, and hence it was not covered in [40], and dealt with later [4, 26, 41]. Note that a Kac-Moody Lie algebra only has finitely many generators and relations, while the Heisenberg algebra has infinitely many. A particular presentation of an algebra should not be fundamental, so it was desirable to have a more intrinsic construction of those representations. More precisely, a definition of an algebra by convolution products is natural, but we would like to understand why we get a particular algebra, namely the affine Lie algebra in our case. We do not have a satisfactory explanation yet. The same applies to Ringel and Lusztig’s constructions.  Δ (α) acting on ∗ n Exercise 3.5.4. [43, Remark 8.19] Define operators P±1 n H (S X) for a (compact) manifold X in a similar way, and check the commutation relation (3.5.2) with r = 1. 3.6. Dimensions of cohomology groups When r = 1, it is known that the gen[∗] [∗] erating function of dimension of HT ((C2 )[d] ) over AT = HT (pt) for d  0 is ∞ 

[∗]

dim HT ((C2 )[d] )qd =

d=0

∞  d=1

1 . 1 − qd

(See [43, Chap. 5].) This is also equal to the character of the Fock space9 of the Heisenberg algebra. Therefore the Heisenberg algebra action produces all coho[∗] mology classes from the vacuum vector |vac = 1(C2 )[0] ∈ HT ((C2 )[0] ). On the other hand we have ∞ ∞   1 [∗]  d d dim HT (U )q = . (3.6.1) r (1 − qd )r d=0

d=1

for general r. Therefore the direct sum of these cohomology groups is not an irreducible representation for the Heisenberg algebra. Let us explain how to see the formula (3.6.1). Consider the torus T action on  d . A framed torsion free sheaf (E, ϕ) is fixed by T if and only if it is a direct U r sum (E1 , ϕ1 ) ⊕ · · · ⊕ (Er , ϕr ) of rank 1 framed torsion free sheaves. Rank 1 framed torsion free sheaves are nothing but ideal sheaves on C2 , hence points in Hilbert 9 The

Δ (α) Fock space is the polynomial ring in infinitely many variables x1 , x2 , . . . . The operators Pn act by either multiplication of xn or differentiation with respect to xn with appropriate constant Δ (α) multiplication. It has the highest weight vector (or the vacuum vector) 1, which is killed by Pn Δ (α) successively to the with n > 0. The Fock space is spanned by vectors given by operators Pn highest weight vector.

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schemes. Thus



 d )T = (U r

(3.6.2)

395

(C2 )[d1 ] × · · · × (C2 )[dr ] .

d1 +···+dr =d

Hence



 d )T ) = HT ((U r [∗]



[∗]

HT ((C2 )[d1 ] × · · · × (C2 )[dr ] )

d1 ,...,dr

d

(3.6.3) =

r  ∞ #

[∗]

HT ((C2 )[di ] ).

i=1 di =0 [∗]  d It is not difficult to show that HT (U r ) is free over AT . (For example, it follows  d and from the vanishing of odd degree nonequivariant cohomology group of U r [∗]  d [∗] d  the spectral sequence relating HT (Ur ) and H (Ur ) ⊗ AT .) Therefore in order to [∗]  d compute the dimension of HT (U r ) over AT , we restrict equivariant cohomology groups to generic points, that is, consider tensor products with the fraction field FT of AT . Then the localization theorem for equivariant cohomology groups gives [∗]  d [∗]  d T an isomorphism between HT (U r ) and HT ((Ur ) ) over FT . Therefore the above observation gives the formula (3.6.1). In view of (3.6.1), we have an action of r copies of the Heisenberg algebra on   [∗]  d T [∗]  d d HT ((Ur ) ) and hence on d HT (Ur ) ⊗AT FT by the localization theorem. It is isomorphic to the tensor product of r copies of the Fock module, so all cohomology classes are produced by the action. This is a good starting point to un [∗]  d derstand d HT (U r ). However this action cannot be defined over non-localized Δ (α) is the ‘diagonal’ Heisenberg in the equivariant cohomology groups. In fact, Pn product, and other non diagonal generators have no description like convolution by Pn .  [∗]  d The correct algebra acting on d HT (Ur ) is the W-algebra W(gl(r)) associated with gl(r). It is the tensor product of the W-algebra W(sl(r)) and the Heisenberg algebra (as the vertex algebra). Its Verma module has the same size as the tensor product of r copies of the Fock module.

Theorem 3.6.4. Operators given by correspondences Pn and multiplication by Chern  [∗]  d classes of the tautological bundle V make d HT (U r ) ⊗AT FT as a W(gl(r))-module, isomorphic to a Verma module. This result is due to Schiffmann-Vasserot [56] and Maulik-Okounkov [36] independently. We will prove this for r = 2 in §6 and obtain a similar result for general G in Theorem 8.4.1. Let us give a comment on the statement, which is a little imprecise. The operators discussed are defined over non-localized equivariant cohomology groups  [∗]  d d HT (Ur ). They and conventional generators of W(gl(r)) are related by an explicit formula, involving elements in C(ε1 , ε2 ) (see (6.4.1) for Virasoro generators). This is one reason why the above theorem is formulated over FT . An explicit formula for generators is known only for sl(r), and will not be reviewed, hence

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an interested reader should read the original papers [36, 56]. Our approach is different. We introduce an A-form of the group W(g) for general g in §8.3, and show that it acts on the non localized equivariant intersection cohomology group in §8.4. 3.7. Intersection cohomology group Recall that the decomposition theorem has a nice form for a semi-small resolution π : M → X:  IC(Xα , χ) ⊗ H[0] (π−1 (xα ))χ , (3.7.1) π∗ (CM ) = α,χ

where we have used the following notation: • CM denotes the shifted constant sheaf CM [dim M]. • IC(Xα , χ) denotes the intersection cohomology complex associated with a simple local system χ on a Zariski open dense subset in the smooth locus of an irreducible closed subvariety Xα of X. (We will simply write IC(Xα ) when χ is trivial. We may also write it IC(X) if Xα is open and dense in X.) • H[0] (π−1 (xα )) is the homology group of the shifted degree 0, which is the usual degree codim Xα . When xα moves in Xα , it forms a local system. H[0] (π−1 (xα ))χ denotes its χ-isotropic component. Exercise 3.7.2. Let Gr(d, r) be the Grassmannian of d-dimensional subspaces in Cr , where 0  d  r. Let M = T ∗ Gr(d, r). Determine X = Spec(C[M]). Study fibers of the affinization morphism π : M → X and show that π is semi-small. Compute graded dimensions of IH∗ of strata, using the well-known computation of Betti numbers of T ∗ Gr(d, r). By Proposition 3.3.1, any fiber π−1 (xα ) of the Gieseker-Uhlenbeck morphism  d → Ud is irreducible. Therefore all the local systems are trivial, and π: U r r   −1 d  2 IC(Bund (3.7.3) π∗ (CU d) = SL(r) ×Sλ (C )) ⊗ C[π (xλ )], r

d=|λ|+d 



d



2 −1 d where xλ denotes a point in the stratum Bund SL(r) ×Sλ (C ), and [π (xλ )] de  −1 d notes the fundamental class of π (xλ ), viewed as an element of H[0] (π−1 (xd λ )). d The main summand is IC(Bund SL(r) ) = IC(Ur ), and other smaller summands could be understood recursively as follows. Let us write the partition λ as (1α1 2α2 . . . ), when 1 appears α1 times, 2 appears α2 times, and so on. We set l(λ) = α1 + α2 + · · · and Stab(λ) = Sα1 × Sα2 × · · · . Note that we have in total l(λ) distinct points in Sλ (C2 ). The group Stab(λ) is the group of symmetries of a configuration in Sλ (C2 ). We have a finite morphism 



2 l(λ) 2 / Stab(λ) → Bund ξ : Ud SL(r) × (C ) SL(r) ×Sλ (C ) 



d 2 2 extending the identity on Bund SL(r) ×Sλ (C ). Then IC(BunSL(r) ×Sλ (C )) is the direct image of IC of the domain. We have the Künneth decomposition for the domain, and the factor (C2 )l(λ) / Stab(λ) is a quotient of a smooth space by a

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finite group. Therefore the IC of the second factor is the (shifted) constant sheaf. We thus have    d 2 ∼ IC(Bund SL(r) ×Sλ (C )) = ξ∗ IC(Ur )  C(C2 )l(λ) / Stab(λ) . Thus [∗]  d HT (U r) =

 d=|λ|+d 

[∗]



[∗]



2 l(λ) IHT (Ud / Stab(λ)) ⊗ C[π−1 (xd r ) ⊗ HT ((C ) λ )].

This decomposition nicely fits with the Heisenberg algebra action. Note that the second and third factors are both 1-dimensional. Thus we have 1-dimensional space for each partition λ. If we take the sum over λ, it has the size of the Fock  [∗]  d  module, and it is indeed the submodule generated by the vector [xd ∅ ] ∈ H (Ur ),  the fundamental class of the point xd ∅ corresponding to the empty partition.  d  d via π.) This statement is a point in Bun , regarded as a point in U (Note xd r r ∅ can be proved by the analysis of the convolution algebra in [16, Chap. 8], but it is intuitively clear as the 1-dimensional space corresponding to λ is the span of Δ (1)α1 P Δ (1)α2 · · · |vac. P−1 −2 The Heisenberg algebra acts trivially on the remaining factor  [∗] IHT (Ud r ). d

The goal of these notes is to see that it is a module of W(sl(r)), and the same is true for ADE groups G, not only for SL(r). Exercise 3.7.4. Show the above assertion that the Heisenberg algebra acts trivially  [∗] on the first factor d IHT (Ud SL(r) ).

4. Stable envelopes The purpose of this lecture is to explain the stable envelope introduced in [∗]  d [∗]  d T [36]. It will nicely explain a relation between HT (U r ) and HT ((Ur ) ). This is what we need to clarify, as we have explained in the previous lecture. The stable envelope also arises in many other situations in geometric representation theory. Therefore we explain it in a wider context, as in the original paper [36]. 4.1. Setting – symplectic resolution Let π : M → X be a resolution of singularities of an affine algebraic variety X. We assume M is symplectic. We suppose that a torus T acts on both M and X so that π is T-equivariant and that the T-action on X is linear. We also assume a technical condition that T contains a C× such that X is a cone with respect to C× and the weight of the symplectic form is positive except in §4.7. (This assumption will be used when we quote a result of the author later.) Let T be a subtorus of T which preserves the symplectic form of M.  d , X = Ud and π is the GiesekerExample 4.1.1. Our basic example is M = U r r Uhlenbeck morphism with the same T, T as above. In fact, we can also take a × × × −1 larger torus T × C× hyp in T, where C hyp ⊂ C × C is given by t → (t, t ).

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Example 4.1.2. Another example is when M = T ∗ (flag variety) = T ∗ (G/B) with X = (nilpotent variety) and π the Grothendieck-Springer resolution. Here T is a maximal torus of G contained in B, and T = T × C× , where C× acts on X by scaling on fibers. We can also consider the same π : M → X as above with smaller T, T .  Let MT be the T -fixed point locus in M. It decomposes as MT = Fα into connected components, and each Fα is a smooth symplectic submanifold of M. Let i : MT → M be the inclusion. We have the pull-back homomorphism  [∗+codim F ] [∗] [∗+codim XT ] α (MT ) = HT (Fα ). i∗ : HT (M) → HT α

Here we take the degree convention as before. Our degree 0 is the usual degree [∗] [∗] dimC M for HT (M), and dimC Fα for HT (Fα ). Since i∗ preserves the usual degree, it shifts our degree by codim Fα . Each Fα has its own codimension, but [∗+codim XT ] (MT ) for brevity. we denote the direct sum as HT The stable envelope we are going to construct goes in the opposite direction [∗] [∗] HT (MT ) → HT (M) and preserves (our) degree.  d , the T and T × C× -fixed point loci are In the above example, M = U r hyp

 d )T = (U r

 d1 × · · · × U  dr , U 1 1

d1 +···+dr =d

 d) (U r

T ×C× hyp

=



{Iλ1 ⊕ · · · ⊕ Iλr },

|λ1 |+···+|λr |

where λi is a partition, and Iλi is the corresponding monomial ideal sheaf (with the induced framing). Also T ∗ (G/B)T = W, where W is the Weyl group. 4.2. Chamber structure Let us consider the space Homgrp (C× , T ) of one parameter subgroups in T , and its real form Homgrp (C× , T ) ⊗Z R. A generic one pa× rameter subgroup ρ satisfies Mρ(C ) = MT . But if ρ is special (the most extreme × case is when ρ is trivial), the fixed point set Mρ(C ) could be larger. This gives us a ‘chamber’ structure on Homgrp (C× , T ) ⊗Z R, where a chamber is a connected component of the complement of the union of hyperplanes given by ρ such that × Mρ(C ) = MT . Exercise 4.2.1. (1) In terms of T -weights on tangent spaces Tp M at various fixed points p ∈ MT , describe the hyperplanes. (2) Show that the chamber structure for M = T ∗ (flag variety) is identified with usual Weyl chambers.  d is identified with the usual (3) Show that the chamber structure for M = U r Weyl chambers for SL(r).

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 d , but with the larger torus (4) Compute the chamber structure for M = U r . T × C× hyp For a chamber C, we have the opposite chamber −C consisting of one parameter subgroups t → ρ(t−1 ) for ρ ∈ C. The stable envelope depends on a choice of a chamber C. 4.3. Attracting set Let C be a chamber and ρ ∈ C. We define the attracting set AX by

 AX = x ∈ X lim ρ(t)x exists . t→0 We similarly define the attracting set AM in M in the same way. As π is proper, we have AM = π−1 (AX ). We put the scheme structure on AM as π−1 (AX ) in these notes. Example 4.3.1. Let X = Ud 2 . In the quiver description, AX consists of closed GL(d)-orbits GL(d)(B1 , B2 , I, J) such that Jf(B1 , B2 )I is upper triangular for any noncommutative monomial f ∈ Cx, y. It is the tensor product variety introduced in [45], denoted by π(Z) therein. As framed sheaves, AM consists of those (E, ϕ) which can be written as exten d1 sions 0 → E1 → E → E2 → 0 (compatible with the framing) for some E1 ∈ U 1  d2 with d = d1 + d2 . and E2 ∈ U 1 We have the following diagram (4.3.2)

Xρ(C

×)

p

j

= XT AX → X, i

where i, j are the natural inclusions, and p is given by AX  x → limt→0 ρ(t)x. If X is a representation of T , XT (resp. AX ) is the 0-weight space (resp. the direct sum of nonnegative weight spaces with respect to ρ(C× )). Hence XT and AX are closed subvarieties, and p is a morphism. The same is true for general X as an affine algebraic variety and the T -action is linear. 4.4. Leaves Let p denote the map AM  x → limt→0 ρ(t)x ∈ MT . This is formally the same as for AX → XT , but it is not continuous, as the limit point may jump at a special point x, as we will see an example below. Nevertheless it is  set-theoretically well-defined. Since MT = Fα , we have the corresponding de −1 composition of AM = p (Fα ). Let Leafα = p−1 (Fα ). By the Bialynicki-Birula theorem ([6], see also [15]), p : Leafα → Fα is a T-equivariant affine bundle. Similarly Leaf− α → Fα denotes the corresponding affine bundle for the opposite chamber −C. (For quiver varieties, they are in fact isomorphic to the vector bundles L± α below. See [45, Prop. 3.14].) Let us consider the restriction of the tangent bundle T M to a fixed point component Fα . It decomposes into weight spaces with respect to ρ:  T (m), T (m) = {v | ρ(t)v = tm v}. (4.4.1) T M|Fα =

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 Then T Leafα = m0 T (m). Note also T (0) = T Fα . Since T preserves the symplectic form, T (m) and T (−m) are dual to each other. From these, one can j×p also check that Leafα −−−→ M × Fα is a Lagrangian embedding. For a later reference, let  def. T (m). (4.4.2) L± α = ±m>0 + − The normal bundle of Fα in Leafα (resp. Leaf− α ) is Lα (resp. Lα ).

Example 4.4.3. Let π : M = T ∗ P1 → X = C2 /±. Let T = C× act on X and C2 / ± 1 T so that it is given by t(z1 , z2 ) mod ± = (tz1 , tz−1 2 ) mod ±. Then X consists of two 1 ∗ 1 points {0, ∞} in the zero section P of T P . If we take the ‘standard’ chamber containing the identity operator, Leaf0 is the zero section P1 minus ∞. On the other hand Leaf∞ is the (strict transform of) the axis z2 = 0. See Figure 4.4.4. For the opposite chamber, Leaf0 is the axis z1 = 0, and Leaf∞ is the zero section minus 0.

Leaf∞

0

∞

Leaf0

Figure 4.4.4. Leaves in T ∗ P1 Definition 4.4.5. We define a partial order  on the index set {α} for the fixed point components so that Leafβ ∩ Fα = ∅ =⇒ α  β. We have ∞  0 in Example 4.4.3. Let AM,α =



Leafβ .

β:βα

Then AM,α is a closed subvariety. We define AM,0

(See [36, Th. 12.4.4].) Next we consider the first Chern class c1 (V) of the tautological bundle over  d . Recall the formula for the coproduct Δ of the Heisenberg operator PΔ (α) U −m 2 Δ (α) = P in Proposition 6.3.1. We have ΔP−m −m (α) ⊗ 1 + 1 ⊗ P−m (α) for rank 2. Δ (α) is given by the The underlying geometric reason for this formula is that P−m Lagrangian correspondence. Note c1 (V) is the fundamental class of the diagonal cut out by c1 (V) as a correspondence. It is not Lagrangian, as it is cut. But one can show that Δc1 (V) − c1 (V) ⊗ 1 − 1 ⊗ c1 (V) involves only the classical r-matrix. (See [36, §10.1.2].) From this together with the above formula for the classical r-matrix, one calculate Δc1 (V) − c1 (V) ⊗ 1 − 1 ⊗ c1 (V). Then we combine it with the formula of c1 (V) ⊗ 1 − 1 ⊗ c1 (V) in Theorem 6.5.1. The formula is given in [36, Th. 14.2.3], but only its restriction to F− is necessary for us.  [∗]  d T Δ ] acts on ∼ FΔ ⊗ F− where HT ((U2 ) ) ⊗AT FT = The commutator [Δc1 (V), Pn loc loc ∼ − the subscript ‘loc’ means ⊗AT FT . We take its restriction to 1 ⊗ F− loc = Floc , com− − − posed with the projection FΔ loc ⊗ Floc → 1 ⊗ Floc . It is an operator on Floc . Let us Δ denote it by [Δc1 (V), Pn ]|F− . Then the formula [36, Th. 14.2.3] gives us loc

Δ ]| − = nL , where L is given by (6.4.1). Theorem 6.6.1. We have [Δc1 (V), Pn n n F loc

Δop c

Since R intertwines Δc1 (V) and 1 (V) by its definition, the above implies that it intertwines Virasoro operators. − Δ intersection of the kernels Note that F− loc is characterized in Floc ⊗ Floc as the  [∗] Δ d of Pm for m > 0. This subspace is nothing but d IHT (USL(2) ) ⊗AT FT by  [∗] d Exercise 3.7.4. Thus d IHT (USL(2) ) ⊗AT FT is a module for the Virasoro algebra, which is the W-algebra associated with g = sl(2). This is the first case of the AGT correspondence mentioned in the Introduction.

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Δ ] is well-defined on non-localized equivariMoreover the operator [Δc1 (V), Pn [∗] Δ Δ Δ Δ ant cohomology IHT (Ud SL(2) ) if Pn = Pn (1) is replaced by ε1 ε2 Pn = Pn ([0]) for n > 0. This is the starting point of our discussion on integral forms. See §8.2.

Remark 6.6.2. For quiver varieties, we have tautological vector bundles Vi associated with vertexes i. The formula for Δc1 (Vi ) is given in the same way (see [36, Th. 10.1.1]). On the other hand, the coproduct of the Yangian Y(g) associated with a finite dimensional simple Lie algebra g has an explicit formula for the first Fourier mode of fields corresponding to Chevalley generators of g. It is given in terms of the classical r-matrix as in the geometric construction, where r is the invariant bilinear form. The constant Fourier modes are primitive, and there is no known explicit formula for the second or higher Fourier modes. When c1 (Vi ) is identified with hi,1 for type ADE quiver varieties, the first Fourier mode of the field corresponding to a Cartan element hi , the geometric and algebraic coproducts coincide. Since hi,1 together with constant modes generates Y(g), two coproducts are equal [37].

7. Perverse sheaves on instanton moduli spaces We now turn to Ud G for general G. 7.1. Hyperbolic restriction on instanton moduli spaces Let ρ : C× → T be a one parameter subgroup. We have associated Levi and parabolic subgroups

 ρ(C× ) , P = g ∈ G lim ρ(t)gρ(t)−1 exists . L=G t→0 ×

Unlike before, here we allow nongeneric ρ so that Gρ(C ) could be different from T . This is not an actual generalization. We can replace T by Z(L)0 , the connected center of L. Then ρ above can be considered as a generic one parameter subgroup in Z(L)0 . We consider the induced C× -action on Ud G . Let us introduce the following notation for the diagram (4.3.2): (7.1.1)

def.

d ρ(C Ud L = (UG )

×)

p

j

def.

d

Ud P = AUd → UG . i

G

Let us explain how this notation can be justified. If we restrict our concern to × the open subscheme Bund G , a framed G-bundle (F, ϕ) is fixed by ρ(C ) if and only if we have an L-reduction FL of F (i.e., F = FL ×L G) so that FL |∞ is sent to ρ(C× ) is the moduli space of framed ∞ × L by the trivialization ρ. Thus (Bund G) 12 L-bundles, which we could write Bund L . The definition of the Uhlenbeck partial compactification is a little delicate, and is known for almost simple groups. Nevertheless it is still known that Ud L is homeomorphic to the Uhlenbeck partial 12 Here

we use the assumption G is of type ADE. The instanton number is defined via an invariant bilinear form on g. For almost simple groups, we normalize it so that the square length of the highest root θ is 2. If G is of type ADE, instanton numbers are preserved for fixed point sets, but not in general. See [12, §2.1].

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compactification for [L, L] when it has only one simple factor ([12, Prop. 4.2.5]), though we do not know whether they are the same as schemes or not. We will actually use this fact later, therefore using the same notation for fixed point subschemes and genuine Uhlenbeck partial compactifications is natural for us. On the notation Ud P : If we have a framed P-bundle (FP , ϕ), the associated framed G-bundle (FP ×P G, ϕ ×P G) is actually a point in the attracting set AUd . G . This Thus the moduli space of framed P-bundles Bund P is an open subset in AUd G d d is why we use the notation Ud P . However a point in UP ∩ BunG does not necessarily come from a framed P-bundle like this. See Exercise 7.1.3 below. Nevertheless we believe that it is safe to use the notation Ud P , as we never consider genuine Uhlenbeck partial compactifications for the parabolic subgroup P. Example 7.1.2. If G = SL(r) and L = S(GL(r1 ) × GL(r2 )), Bund L is the moduli space of pairs of framed vector bundles (E1 , ϕ1 ) and (E2 , ϕ2 ). On the other hand, Bund P is the moduli space of vector bundles E which can be realized as an extension 0 → E1 → E → E2 → 0. (In this situation, one can show that E determines E1 and E2 .) Exercise 7.1.3 (cf. an example in [12, §4.4]). Consider the case G = SL(r). Suppose (E, ϕ) is a framed vector bundle and that 0 → E1 → E → E2 → 0 is an an exact sequence compatible with the framing. Here we merely assume E1 and E2 are torsion-free sheaves. Are E1 and E2 locally free? Give a counter-example. d b d Definition 7.1.4. The hyperbolic restriction functor p∗ j! : Db T (UG ) → DT (UL ) will be denoted by ΦP L,G , or, if the groups are clear from the context, simply by Φ.

We have the following associativity of hyperbolic restrictions. Proposition 7.1.5. Let Q be another parabolic subgroup of G, contained in P and let M denote its Levi subgroup. Let QL be the image of Q in L and we identify M with the corresponding Levi group. Then we have a natural isomorphism of functors P L ∼ Q ΦQ M,L ◦ ΦL,G = ΦM,G .

Proof. It is enough to show that d d Ud P ×Ud UQL = UQ . L

This is easy to check. See [12, Prop. 4.5.1].



7.2. Exactness For a partition λ, let Sλ C2 be a stratum of a symmetric product as before. Let Stab(λ) = Sα1 × Sα2 × . . . if we write λ = (1α1 2α2 . . . ). We consider an associated covering (C2 )α1 × (C2 )α2 × · · · \ diagonal → Sλ (C2 ). Let ρ be a simple local system over Sλ C2 corresponding to an irreducible representation of Stab(λ). We consider the following class of perverse sheaves:

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Definition 7.2.1. Let Perv(Ud L ) be the additive subcategory of the abelian category of semisimple perverse sheaves on Ud L , consisting of finite direct sum of  2 ), 1  ρ) for various d  , λ, ρ. ×S (C IC(Bund λ L Here we consider the stratification of Ud L as in (2.2.2):  2 Bund Ud L ×Sλ (C ). L = d=|λ|+d 

It is the restriction of the stratification (2.2.2) to Ud L. Let us explain why we need to consider nontrivial local systems, even though d our primary interest will be on IC(Ud G ): When we analyze IC(UG ) through the hyperbolic restriction functor, IC sheaves for nontrivial local systems occur. This phenomenon can be seen for type A as follows.  d and consider the hyperbolic restriction for Let us take the Gieseker space U r d T ∼ T a chamber C for the T -action. We have p∗ j! π∗ (CU  d ) = π∗ (C(U  d )T ), where (Ur ) r r  d )T → (Ud )T is the restriction of π by Corolis the fixed point set and πT : (U r r lary 5.4.2(2). The fixed point sets are given by Hilbert schemes and symmetric products, and πT factors as  d )T = (C2 )[d1 ] × · · · × (C2 )[dr ] (U r d1 +···+dr =d π×···×π

−−−−−−→



κ

T S d1 C 2 × · · · × S dr C 2 − → Sd C2 = (Ud r) ,

where κ is the ‘sum map’, defined by κ(C1 , · · · , Cr ) = C1 + · · · + Cr if we use the ‘sum notation’ for points in symmetric products, like x1 + x2 + · · · + xd . The pushforward for the first factor π × · · · × π is  CS (C2 )  · · ·  CS r (C2 ) |λ1 |+···+|λr |=d

λ1

λ

by the discussion in §3.7, where λ1 , . . . , λr are partitions. Therefore we do not have nontrivial local systems. But κ produces nontrivial local systems. Since κ is a finite morphism, in order to calculate its pushforward, we need to study how κ restricts to covering on strata. For example, for λ = (1r ), d1 = d2 = · · · = dr = 1, it is a standard Sr -covering (C2 )r \ diagonal → S(1r ) (C2 ). We have the following d d Theorem 7.2.2. ΦP L,G sends Perv(UG ) to Perv(UL ). d By the remark after Theorem 5.3.1 we know that ΦP L,G sends Perv(UG ) to semisimple complexes. From the factorization, it is more or less clear that these must be direct sums of shifts of simple perverse sheaves in Perv(Ud L ). Therefore the actual content of this theorem is the t-exactness, that is shifts are unnecessary. For type A, it is a consequence of Corollary 5.4.2. For general G, we use hyperbolic semi-smallness (Theorem 5.5.3). The detail is given in [12, Appendix A]. We sketch another shorter proof added in revision,

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that was suggested by a referee of [12]. By the recursive nature and the factorization property of instanton moduli spaces, it is enough to estimate dimension of the extreme fibers, i.e., p−1 (d · 0), p−1 − (d · 0) in (5.5.2). Considering the stratification (2.2.1), we see that enough to estimate dimensions of their intersections with Bund G . Furthermore using the associativity of hyperbolic restriction (Proposition 7.1.5), it is enough to prove the case L = T . In fact, we can further reduce to the case of the hyperbolic restriction for the larger torus T × C× hyp . The associativity (Proposition 7.1.5) remains true for× the larger torus. Moreover the fixed C point set is the single point d · 0 as (C2 ) hyp = {0}. We then consider the case  d are G = GL(r). The attracting and repelling sets in the symplectic resolution U r Lagrangian, as we explained in §4.4. For general G, we take a faithful embedding d G → GL(r). We have the induced embedding Bund G → BunGL(r) which respects symplectic structures. Therefore the intersections of p−1 (d · 0), p−1 − (d · 0) with are isotropic, hence at most half dimensional. Bund G 7.3. Calculation of the hyperbolic restriction Our next task is to compute the d direct summands of ΦP L,G (IC(UG )). The two most extreme simple ones are: (a) IC(Ud L ), (b) CS(d) (C2 ) . The other direct summands are basically products of type (a) and (b). Let us first consider (a). Let us restrict the diagram (7.1.1) to Bund L . Note that a point in d −1 p (BunL ) is a genuine bundle, cannot have singularities, as singularities are preserved or increased under p. Thus the diagram sits in moduli spaces of genuine bundles as p d j d Bund L BunP → BunG . i

Bund P

Then is a vector bundle over Bund L . We can verify this claim as follows. d 1 2 For F ∈ BunG , the tangent space TF Bund G is H (P , gF (−∞ )), where gF is the d associated bundle F ×G g. If F ∈ BunL , we have the decomposition ∼ H1 (P2 , lF (−∞ )) ⊕ H1 (P2 , n+ (−∞ )) ⊕ H1 (P2 , n− (−∞ )), H1 (P2 , gF (−∞ )) = F F according to the decomposition g = l ⊕ n+ ⊕ n− . Here n+ is the nilradical of p, and n− is the nilradical of the opposite parabolic. The first summand gives the tangent bundle of Bund L . Thus the normal bundle is given by sum of the second and third summands. Moreover, the second and third summands are Leaf and Leaf− respectively. ∼ C d . It extends to a Therefore we have the Thom isomorphism p∗ j! CBund = BunL G canonical isomorphism ! d ∼ Hom(IC(Bund L ), p∗ j IC(UG )) = C.

In other words, p−1 (yβ ) ∩ X0 in Theorem 5.5.3 is the fiber of the vector bundle d Bund P → BunL , hence we have the canonical isomorphism given by its fundamen∼ C. tal class, Hdim X−dim Yβ (p−1 (yβ ∩ X0 )χ =

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On the other hand, the multiplicity of the direct summand CS(d) (C2 ) of type (b) has no simple description. It is the top degree homology group of the space p−1 (d · 0) ∩ Bund G by Theorem 5.5.3, hence has a base given by irreducible components. But it is hard to calculate the base so we introduce the following space: def.

P d Ud ≡ Ud L,G = Hom(CS(d) (C2 ) , ΦL,G (IC(UG ))). d Thus Ud ⊗ CS(d) (C2 ) is the isotropic component of ΦP L,G (IC(UG )) for CS(d) (C2 ) . From the factorization, we have the canonical isomorphism   d 1 ⊗α1 2 ∼ IC(Bund ⊗ (U2 )⊗α2 ⊗ · · · ), ΦP L,G (IC(UG )) = L ×Sλ (C ), (U ) d=|λ|+d 

where λ = (1α1 2α2 . . . ) and we view (U1 )⊗α1 ⊗ (U2 )⊗α2 ⊗ · · · as a representation of Stab(λ) = Sα1 × Sα2 × · · · . Though (U1 )⊗α1 ⊗ (U2 )⊗α2 ⊗ · · · may not be irreducible, it is always semisimple. We understand the associated IC-sheaf as the direct sum of IC-sheaves of simple constituents. If we take the global cohomology group, contributions of nontrivial local systems vanish. (See [12, Lemma 4.8.7].) Therefore (after taking an isomorphism ∼ AT ) we get H∗T (Sλ C2 ) =     d 1 2 ∼ H∗T (ΦP H∗T (IC(Bund L,G (IC(UG )) = L )) ⊗C Sym U ⊕ U ⊕ · · · , (7.3.1)

d



d

  d 1 2 ∼ H∗T (ΦB T ,G (IC(UG )) = AT ⊗C Sym U ⊕ U ⊕ · · · ,

d

where the lower isomorphism is the special case of the upper one as Bund T = ∅ for d = 0 and is a point for d = 0. The first result on Ud is Lemma 7.3.2 ([12, Lemma 4.8.11]). dim Ud = rank G − rank[L, L], in particular dim Ud = rank G if L = T . This result was proved as (1) a reduction to the case L = T via the associativity (Proposition 7.1.5), (2) a reduction to a computation of the ordinary restriction to S(d) A2 thanks to a theorem of Laumon [30], and (3) the computation of the ordinary restriction to S(d) A2 in Theorem 7.10 of [11]. Instead of explaining the detail of this argument, let us give two explanations of dim Ud = rank G for L = T in the next two subsections. In fact, we will also ∼ h, the Cartan subalgebra of G. (This will see that there is an isomorphism Ud = be actually one of our goals.)

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7.4. Hyperbolic restriction for Gieseker partial compactification Let us suppose G = SL(r) and consider the hyperbolic restriction of π∗ (CU  d ). Let us supr pose L = T for brevity. We introduce def.

V d = Hom(CS(d) (C2 ) , ΦB  d ))) T ,G (π∗ (CU r

as an analog of Ud . As in (7.3.1), we have    1 2 ∼ H∗T (ΦB (π (C ))) A ⊗ Sym V ⊕ V ⊕ · · · . = ∗ T C d T ,G U d

r

∼ T By the stable envelope, we have an isomorphism ΦB  d )) = π∗ (C(U  d )T ) T ,G (π∗ (CU r r T (see Corollary 5.4.2). We decompose π∗ (C(U  d )T ) according to the the description r  d )T . One can easily check that a direct summand has nonzero con(3.6.2) of (U r tribution to V d only when one of d1 , . . . , dr is 1 and the rest are zero. More, π∗ (C(C2 )[d] )) is naturally isomorphic to the fundamental class over Hom(CS (d)(C2 ) [π−1 (d · 0)] of the fiber of π at d · 0 ∈ S(d) (C2 ). In particular, we have a base {e1 , . . . , er } corresponding to summands (d1 , . . . , dr ) = (1, 0, . . . , 0), (0, 1, 0, . . . , 0) up to (0, . . . , 0, 1), and V d is r-dimensional. Moreover in the decomposition of π∗ (CU  d ) in (3.7.3), a direct summand conr tributes to V d only if d  = d (and λ = ∅) or λ = (d) (and d  = 0). Therefore we have a natural direct sum decomposition V d = Ud ⊕ C[π−1 (d · 0)], where d · 0 is considered as a point in the stratum Bun0SL(r) ×S(d) (C2 ). In particular, we obtain dim Ud = r − 1 = rank SL(r) as expected. Δ ([0])|vac = ε ε P Δ |vac. By PropoNote also that the class [π−1 (d · 0)] is P−d 1 2 −d sition 6.3.1 it is equal to e1 + · · · + er . Since Ud corresponds to the subspace killed by the Heisenberg operators, we also see 

∼ a1 e1 + · · · + ar er ai = 0 . Ud = Thus Ud is isomorphic to the Cartan subalgebra of sl(r). Furthermore for r = 2, − 1 2 we have F− loc = FT ⊗C Sym(U ⊕ U ⊕ · · · ), where Floc is the Fock space of the (1) (2) − = P Heisenberg operator Pn n − Pn considered in the previous lecture. In fact, − , as [π−1 (d · 0)] is a it will be naturally considered as the Fock space for ε1 ε2 Pn cohomology class with compact support, when we will consider integral forms in §8. 7.5. Affine Grassmannian for an affine Kac-Moody group In this subsection, we drop the assumption that G is of type ADE. Recall that it is proposed in [10] that G-instanton moduli spaces on R4 /Z play the role of the affine Grassmannian for an affine Kac-Moody group, as we mentioned in the Introduction. Let us give an interpretation of dim Ud = rank G in this framework. We restrict ourselves to the case  = 1. The main conjecture in [10] is proved in this case. Moreover there are no differences in instanton moduli spaces for

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various G with a common Lie algebra over R4 . Therefore we expect that corresponding representations are of the Langlands dual affine Lie algebra. Let us denote it by g∨ aff . If G is of type ADE, then it is the untwisted affine Lie algebra of Lie G. If G is of type BCFG, it is the Langlands dual of the untwisted affine Lie algebra of Lie G, hence the notation is reasonable. Here the Langlands dual is just given by reversing arrows in the Dynkin diagram. Concretely it is a twisted affine Lie algebra given by the following table: the affine Lie algebra g∨ aff

G

(1)

Xr (X = ADE)

Xr

Br

A2r

Cr

Dr+1

F4

E6

G2

D4

(2)

(2)

(2) (3)

Here we follow [27] for the notation of Lie algebras. Since we are considering  = 1, our Ud G should correspond to a level 1 rep∨ resentation of gaff . It is known that all level 1 representations are related by (1) Dynkin diagram automorphisms for our choice of g∨ aff (e.g., rotations for Ar ) [27, (12.4.5)]. Therefore we could just take the 0th fundamental weight Λ0 without loss of generality. In fact, we do not have a choice of a highest weight in Ud G, so the geometric Satake correspondence could not make sense otherwise. Let us consider L(Λ0 ), the irreducible representation of g∨ aff with highest weight d Λ0 . The weight corresponding to UG is Λ0 − dδ. We only have the discrete parameter d in the geometric side. Fortunately all dominant weights of L(Λ0 ) are of this form, and geometric Satake for affine Kac-Moody groups was formulated only for dominant weights when [10] was written. By [27, Prop. 12.13] we have multL(Λ0 ) (Λ0 − dδ) = pA (d), where



pA (d)qd =

d0



(1 − qn )− mult nδ .

n1

∨ Moreover mult nδ = r if g∨ aff = Xr for type ADE. Next suppose gaff is the Lang(1) ∨ lands dual of Xr where X is of type BCFG. Let r be the lacing number of g∨ aff , i.e., the maximum number of edges connecting two vertexes of the Dynkin dia(1) ∨ gram of g∨ aff (and Xr ). Then mult nδ = r if n is a multiple of r , and equals the number of long simple roots in the finite dimensional Lie algebra Xr otherwise. Explicitly r − 1 for Br , 1 for Cr and G2 , and 2 for F4 . See [27, Cor. 8.3]. Let us turn to the geometric side. As we have explained in §5.6, the hyperbolic restriction with respect to a maximal torus corresponds to a weight space. Our T is a maximal torus of G, but not of the affine Kac-Moody group. So let us × −1 consider a larger torus T = T × C× hyp , where the second C hyp is a subgroup (t, t ) (1)

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of C× × C× acting on C2 preserving the symplectic form. There is only one fixed  point (for each Ud G ) with respect to T , as we have remarked at the end of §7.2.  It is hyperbolic Let us denote the corresponding hyperbolic restriction by Φ.  semi-small. (In fact, we proved the property first for Φ, as we mentioned before.) Therefore we expect d  Φ(IC(U G )) is naturally isomorphic to a weight space of a level 1 representation of g∨ aff . In our case, we have already computed this space implicitly in §7.3 for type ADE. It is equal to the degree d part of the right hand side of (7.3.1). Otherwise solve Exercise 5.5.4. When G is of type BCFG, we have a different behavior because of the mismatch of instanton numbers under the restriction, explained in footnote 5. One can check that it matches with the above character formula. For type ADE, the representation L(Λ0 ) has an explicit realization by vertex operators (Frenkel-Kac construction) [27, §14.8]. The underlying vector space is given by  z−d ⊗ h) ⊗ C[Q], Sym( d>0

where h (resp. Q) is the Cartan subalgebra (resp. the root lattice) of Lie G. Therefore it is natural to identify Ud with z−d ⊗ h. Let us remark that a conjecture (which is a theorem for  = 1) in [10] involves d  IH∗ (Ud G ) instead of Φ(IC(UG )). Two spaces have the same dimension thanks to [30] mentioned above, but the information of the grading in IH∗ (Ud G ) is lost in d  Φ(IC(UG )). Therefore our formulation here is simpler than [10]. 7.6. Heisenberg algebra representation on localized equivariant cohomology Let us fix a Borel B, and consider the parabolic subgroups Pi ⊃ B for each vertex i of the Dynkin diagram of G. Let Li be the Levi factor. By the associativity of the hyperbolic restriction (Proposition 7.1.5), we have a natural decomposition d ∼ d Ud T ,G = UT ,Li ⊕ ULi ,G .

∼ SL(2), we apply the construction in §7.4 to get a Fock space Since [Li , Li ] = representation on FT ⊗C Sym(U1T ,Li ⊕ U2T ,Li ⊕ . . . ). Let us denote the Heisenberg i . We extend it to F ⊗ Sym(U1 2 operator by Pm T C T ,G ⊕ UT ,G ⊕ . . . ) by letting it act trivially on Ud Li ,G . So we have rank G copies of Heisenberg generators acting on FT ⊗C Sym(U1T ,G ⊕ U2T ,G ⊕ . . . ). Proposition 7.6.1. We have 1 , ε1 ε2 where αi is the simple root of G corresponding to Pi . Therefore    d 1 2 ∼ F H∗T (ΦB (IC(U ))) ⊗ F ⊗ Sym U ⊕ U ⊕ · · · (7.6.3) = AT T T C T ,G G T ,G T ,G i j [Pm , Pn ] = −mδm,−n (αi , αj )

(7.6.2)

d

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is naturally the Fock representation of the Heisenberg algebra associated with the Cartan subalgebra h of g = Lie G. i , P j ]. If i = j, it is the SL(2)-case. Proof. Let us consider the commutator [Pm n (1) (2) (1) (2) Hence it is just given by [Pm − Pm , Pn − Pn ] in the notation in §6.4. For two distinct vertexes i, j, we consider the corresponding parabolic subd ∼ d group Pi,j with Levi factor Li,j . We have Ud T ,G = UT ,Li,j ⊕ ULi,j ,G . Since [Li,j , Li,j ] is either SL(3) or SL(2) × SL(2). For SL(2) × SL(2), the commutator is clearly  [∗]  d T 0. For SL(3), we consider HT ((U3 ) ) as the tensor product of three copies of (1) (2) (3) Fock space as in §6.4. Then we have three Heisenberg operators Pm , Pm and Pm Δ = P (1) + P (2) + P (3) . It is also clear that P i = P (1) − P (2) , P j = P (2) − P (3) and Pm m m m m m m m m m by a consideration of the associativity of the hyperbolic restriction. Hence we get (7.6.2). In fact, there is a delicate point here. We need to choose a polarization for instanton moduli spaces so that it is compatible with a polar d for each Li,j . See [12, §6.2] for detail. ization for U 3 Finally the size of (7.6.3) is the same as that of the Fock representation by Lemma 7.3.2. Since the Fock representation is irreducible, we conclude (7.6.3) is the Fock representation. 

8. W-algebra representation on



∗ d d IHT (UG )

The goal of this final lecture is to explain the construction of a W-algebra  representation on d IH∗T (Ud G ). In fact, if we assume the level is generic, or take ⊗AT FT in equivariant cohomology, we have already achieved it. This is because the W-algebra is known to be a (vertex) subalgebra of the Heisenberg (vertex) algebra at generic level by Feigin-Frenkel (see [21, Ch. 15]). So we just restrict the  d Heisenberg algebra representation on d H∗T (ΦB T ,G (IC(UG ))) to W, and use the ∗ B d ∗ ∼ localization theorem HT (ΦT ,G (IC(UG ))) ⊗AT FT = IHT (Ud G ) ⊗AT FT . But this is not an interesting assertion, as it does not explain any geometric meaning of the W-algebra. One way to explain such a meaning is to prove the Whittaker conditions are 0 d satisfied for fundamental classes [Ud G ] ∈ IHT (UG ) as in [12, §8]. We will go half way towards this goal, namely we will explain a construction of a representation  ∗ d of an integral form of the W-algebra on d IHT (UG ). The remaining half re(κ) quires an introduction of generating fields (denoted by Wi in [21] and by W in [12]), which is purely algebraic, and hence is different from the theme of these lectures. We think that the integral form itself is an important object, as we can recover arbitrary level (and highest weight) representations by a specialization AT → C. Thus highest weights and level are identified with equivariant variables. We plan to discuss this further in the future. The W-algebra is defined by the quantum Drinfeld-Sokolov reduction, that is the cohomology of the so-called BRST complex associated with an affine Lie

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algebra [21, Ch. 15]. This definition is algebraic, and will not be recalled here. The integral form mentioned above is also defined by the BRST complex. Therefore we will treat its characterization as a subalgebra of the Heisenberg algebra (Theorem 8.3.1) as a black box, and regard it as defined in this way. 8.1. Four types of nonlocalized equivariant cohomology groups We consider cohomology groups with both compact and arbitrary support and having coefd ficients in IC(Ud G ) or its hyperbolic restriction ΦT ,G (IC(UG )). So we consider four types of nonlocalized equivariant cohomology groups. They are related by natural adjunction homomorphisms as   d IH∗T,c (Ud H∗T,c (ΦB G) → T ,G (IC(UG )))   d d (8.1.1) d H∗T (ΦB IH∗T (Ud → T ,G (IC(UG ))) → G ). d

d

(See (5.3.3).) One can show that they are free AT -modules, and the homomorphisms are inclusions, which become isomorphisms over FT ([12, §6]). Moreover, we have a natural Poincaré pairing between the first and fourth cohomology groups. When we compare it with Kac-Shapovalov form so that it is a pairing between the Verma module with highest weight a and the dual of the Verma module with highest weight −a, we need to make a certain twist of this pairing. In particular, the pairing is sesquilinear in equivariant variables. See [12, §6.8]. Let us ignore this point, as we will not discuss details of highest weights. The pairing between the second and third can be defined if we replace one of B− , since by Theorem 5.3.1, the hyperbolic restrictions by the opposite one ΦT ,G ! ∗ ! Dp∗ j = p! j D = p−∗ j− . This is compensated by the twist above, hence we have a pairing between the second and third. i = ε1 ε2 Pi = Pi ([0]) be the Heisenberg operator associated with the Let P n n n fundamental class of the origin 0 ∈ C2 . This operator is always well-defined on the middle two cohomology groups in (8.1.1). It satisfies i i , P [P m n ] = −mδm,−n (αi , αj )ε1 ε2 . Note that the right hand side is a polynomial in ε1 , ε2 contrary to (7.6.2). The pairing is invariant. def. i with Let HeisA (h) denote the algebra over A = C[ε1 , ε2 ] generated by P n the above relations. It is an A-form of the Heisenberg algebra. Two middle cohomology groups in (8.1.1) are HeisA (h)-modules dual to each other. ∼ AT , Let us look at (7.3.1). The isomorphism uses the identification H∗T (Sλ C2 ) = i 2  which is given by the fundamental class [Sλ C ]. It is not compatible with Pn , but ∼ AT is since it is given by the fundamental class [0]. Therefore we H∗T,c (Sλ C2 ) = should consider the cohomology group with compact support. We get

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 ∗ B d Proposition 8.1.2. d HT,c (ΦT ,G (IC(UG ))) is a highest weight module of HeisA (h) with the highest weight vector |vac, i.e.,  d ∼ H∗T,c (ΦB T ,G (IC(UG ))) = HeisA (h)|vac. d

8.2. Integral form of Virasoro algebra Consider the case G = SL(2). Recall that we can normalize Virasoro operators so that they act on non-localized equivariant cohomology as we have discussed after Theorem 6.6.1. More precisely, as we set n = ε1 ε2 Ln . Then it is a well-defined i = ε1 ε2 Pi , it is natural to introduce L P n n ∗ d ∗ d operator on IHT (USL(2) ) and IHT,c (USL(2) ) by Theorem 6.6.1. The relation (6.4.2) is modified to 

  m3 − m . Ln ] = ε1 ε2 (m − n) Lm+n + ε1 ε2 + 6(ε1 + ε2 )2 δm,−n [ Lm ,  12 The relation is defined over C[ε1 , ε2 ]. Therefore we have the integral form of the Virasoro algebra defined over C[ε1 , ε2 ]. Let us denote it by VirA . We have an embedding VirA ⊂ HeisA (hsl2 ), where hsl2 is the Cartan subalgebra of sl2 . Proposition 8.2.1. IH∗T,c (Ud SL(2) ) is a highest weight module with the highest weight vector |vac. We already know that |vac is killed by  Ln . The highest weight is computed ∗ in (6.4.3). Hence we need to check IHT,c (Ud SL(2) ) = VirA |vac. This is done by comparing graded dimensions of both sides ([12, Prop. 8.1.11]). (This argument works for general G.) We call IH∗T,c (Ud SL(2) ) the universal Verma module, as it specializes to the Verma module in the usual sense under a specialization AT → C. Then IH∗T (Ud SL(2) ) is the dual universal Verma module. 8.3. Integral form of W-algebra Let us take the parabolic subgroup Pi and its Levi factor Li as in §7.6. Then we have the corresponding integral Virasoro algebra as a subalgebra in HeisA (h). Let us denote it by Viri,A . Let HeisA (α⊥ i ) denote the integral Heisenberg algebra for the root hyperplane αi = 0 corresponding to Li . Then Viri,A ⊗A HeisA (α⊥ i ) is an A-subalgebra of HeisA (h). More precisely we need to consider them as vertex algebras, but we ignore this point. Then an integral version of Feigin-Frenkel’s result is Theorem 8.3.1 ([12, Th. B.6.1]). Let WA (g) be an A-form of the W-algebra defined by an A-form of the BRST complex. Then

∼ Viri,A ⊗A HeisA (α⊥ WA (g) = i ), i

where the intersection is taken in HeisA (h). The original version states that the same result is true for generic level, and can be deduced from the version above by taking ⊗A C(ε1 , ε2 ). The level k in the

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usual approach is related to ε1 , ε2 by the formula ε2 k + h∨ = − , ε1 where h∨ is the dual Coxeter number of g = Lie G. As mentioned above, we will not review the definition of the BRST complex here. For our purpose, it is enough to consider the right hand side of the isomorphism in Theorem 8.3.1 as a definition of WA (g). Remark 8.3.2. Two equivariant variables ε1 and ε2 are symmetric in the geometric context. However, ε1 and ε2 play very different roles in the A-form on the BRST complex above. On the other hand, VirA and HeisA both possess the symmetry ε1 ↔ ε2 . Therefore the theorem implies the symmetry on WA (g). This symmetry is nontrivial, and is called the Langlands duality of the W-algebra. (If g is not of type ADE, g must be replaced by its Langlands dual, as the Heisenberg commutation relation involves the inner product (αi , αj ).) See [21, Ch. 15]. Note also the symmetry is apparent in the geometric context, but only for type ADE, and is not clear for other types. (To realize WA (g), one should use instanton moduli spaces for twisted affine Lie algebras. Then the roles of ε1 and ε2 are asymmetric.) 8.4. Intersection cohomology with compact support as universal Verma module of the W-algebra We use the associativity of the hyperbolic restriction as in §7.6. Combining Theorem 8.3.1 with the result in §8.2, we find that  d i H∗T,c (ΦP Li ,G (IC(UG ))) d

 d is a module of WA (g). Here the intersection is taken in d H∗T,c (ΦB T ,G (IC(UG ))). One can check that this intersection is equal to IH∗T,c (Ud G ). (See the proof of [12, Prop. 8.1.7].) Therefore  ∗ d Theorem 8.4.1 ([12, §8.1]). d IHT,c (UG ) is a WA (g)-module. It is a highest weight module with the highest weight vector |vac. We do not review the highest weight, but it is given by an explicit formula  in equivariant variables as in (6.4.3). We call d IH∗T,c (Ud G ) the universal Verma module as for the Virasoro algebra.

9. Concluding remarks 9.1. AGT correspondence As mentioned in Introduction, the AGT correspondence is based on a hypothetical 6-dimensional quantum field theory. The existence of this theory is rather difficult to justify even at a physical level of rigor, as its Lagrangian description is unknown. There are surveys on the AGT correspondences and other related subjects by a group of physicists [60], and another by Tachikawa written for mathematicians [59]. Let us try to give a very short

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summary of what the author has understood from [59]. We will also omit several important points, say topological twists, to make it of reasonable length, hence the reader should read it with care, and consult [59] for corrections. The 6-dimensional quantum field theory is associated with a Dynkin diagram Γ of type ADE. Let us denote it by SΓ . It gives a number ZSΓ (X6 ) for a compact Riemannian 6-dimensional manifold X, called the partition function. If X6 has a boundary ∂X6 , ZSΓ (X6 ) depends on a Riemannian metrics on ∂X6 . Hence ZSΓ (X6 ) is a function on the space of Riemannian metric on ∂X6 . It is not an arbitrary function, and is an element of the space of states or quantum Hilbert space ZSΓ (Y 5 ) associated with a compact 5-manifold Y 5 = ∂X6 . The reader should be familiar with Atiyah’s axiomatic approach to topological quantum field theories, and the theory SΓ should rise a similar structure. A function may be multi-valued, or a section of a line bundle, but we ignore this point, as we will consider it as an element of ZSΓ (Y 5 ). For an application to our study of instanton moduli spaces, we take R4 × C as a 6-manifold X, where C is a Riemann surface. We endow R4 with the C× × C× action as in the main body. It can be regarded as a ‘skeleton’ of a Riemannian metric on R4 appeared above. In particular, the partition function ZSΓ (R4 × C) depends on the equivariant variables ε1 and ε2 . On the other hand, as a function on C, ZSΓ (R4 × C) depends only on a conformal structure on C, hence C is a Riemann surface instead of a Riemannian 2-manifold. We view ZSΓ (R4 × C) as a partition function ZSΓ [R4 ] (C) for a 2-dimensional conformal field theory SΓ [R4 ]. The latter SΓ [R4 ] is called the dimension reduction of SΓ by R4 , and physicists accept that this procedure from the 6d theory to a 2d theory is possible. Care is required as R4 is noncompact. We need to specify a point in the moduli space of vacua, or rather, the partition function is a function on the moduli space of vacua which is a finite dimensional manifold associated with a 4-dimensional quantum field theory SΓ [C] obtained as the dimension reduction of SΓ by C. Unfortunately there is no mathematically rigorous definition of the moduli space of vacua for an arbitrary quantum field theory, but for SΓ [C], it is believed to be an affine space, something like the base of Hitchin integrable system on C associated with the group G13 whose Dynkin diagram is Γ , or its modification. It should also be emphasized that the partition function may have poles, and is defined only locally on the moduli space of vacua. When C has punctures, fields are allowed to have singularities there. We need to specify the type of these singularities, as is explained in [59, §3.5]. It very roughly corresponds to one for the moduli space of Higgs bundles. Since the Hitchin integrable system appears as the moduli space of vacua, this sounds reasonable. But Lusztig-Spaltenstein duality is involved, and the actual relation is quite far from simple. We will not go further detail. 13 Tachikawa

told me that we need to specify a group upon a reduction.

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As a 2-dimensional quantum field theory, SΓ [R4 ] has the space of states associated with S1 . A fundamental claim is  IH∗T ×C× ×C× (Ud ZSΓ [R4 ] (S1 ) = ZSΓ (R4 × S1 ) = G ) ⊗ FT . d

In other words, the direct product of localized equivariant intersection cohomology groups of instanton moduli spaces is the space of states of SΓ [R4 ]. Its element  . The latter is related is a function in equivariant variables ε1 andε2 as well as a with the moduli space of vacua above. Let us take an elliptic curve Eτ of period τ as an example of C. Since Eτ is obtained by a gluing of a cylinder, we have ZSΓ [R4 ] (Eτ ) = trZ

SΓ [R4 ]

(S1 ) (A),

where A is an operator on ZSΓ [R4 ] (S1 ) corresponding to the cylinder. Note that the base of Hitchin integrable system on Eτ is regarded as a single point, hence in the moduli space of ZSΓ [R4 ] (Eτ ) does not have an extra dependence on points √ 2π −1τd on the summand vacua. It is believed that the operator A is given by e  d ). Thus by §7.3 the trace is equal to (U (1 − qn )− rank G with IH∗T ×C×√ n1 G ×C× q = e2π −1τ . This is essentially a power of the Dedekind eta function η, which √ satisfies η(−1/τ) = (− −1τ)1/2 η(τ). This is compatible with the considerations above, as Eτ and E−1/τ are isomorphic. In fact, the dimension reduction of SΓ by Eτ is known to be the 4-dimensional N = 4 supersymmetric gauge theory, whose topological (Vafa-Witten) twist roughly gives generating functions of Euler numbers of instanton moduli spaces [61]. The conjecture that the generating ∼ E−1/τ . function is a modular form is explained as a consequence of Eτ =  d The sum of fundamental cycles d [UG ] is a vector in ZSΓ [R4 ] (S1 ). It is believed that it is ZSΓ [R4 ] (C) where C is the unit disk with an irregular puncture at the origin. For type Ar−1 , we can consider the rth power e(V)r of the equivariant Euler  d . It is believed that the correspondclass of the tautological vector bundle V on U r ing C is the unit disk with two regular punctures. Since V is of rank d, e(V)r has  d)  d . Hence e(V)r lives in H[0] × × (U degree 2dr, which is the dimension of U r r T ×C ×C in our convention, hence the rth power is expected to be natural. We also observe  d. , Ed (−∞ )), where Ed is the universal bundle over P2 × U that V⊕r is Ext1 (O⊕r r P2 is the special case of E with d = 0, we can more generally Considering O⊕r d 2 P d × U  d . This class for r = 1 is studied consider the class Ext• (Ed  , Ed (−∞ )) on U r r in [14]. See also [53]. The corresponding C is a cylinder with a regular puncture. We do not know how to generalize these constructions outside type A. Now comes the punch line of this story. The AGT correspondence predicts that ZSΓ [R4 ] is the conformal field theory associated with the W-algebra for Γ . This is an almost mathematically rigorous statement, except that only the chiral part of a conformal field theory is usually studied as a vertex algebra in the

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References

mathematical community. Nevertheless we can still derive rigorous statements, for example our result that ZSΓ [R4 ] [S1 ] is the Verma module of W is the very first  consequence. The vector d [Ud G ] = ZSΓ [R4 ] (C) is a characterization in terms of W, i.e., it is the Whittaker vector. For further results, besides the papers [14, 53] mentioned above, there are many papers in physics literature for type A, and also for classical groups. See Tachikawa’s survey in [60]. 9.2. Post-requisite Further questions and open problems are listed in [12, §1.11]. In order to do research in those directions, the following would be necessary in addition to what is explained in this lecture series. • As explained above, the AGT correspondence predicts that ZSΓ [R4 ] is the conformal field theory associated with the W-algebra. In order to understand this in a mathematically rigorous way, one certainly needs to know the theory of vertex algebras (e.g., [21]). In fact, we still lack a fundamental understanding why equivariant intersection cohomology groups of instanton moduli spaces have structures of vertex algebras. We want to have an intrinsic explanation without any computation, like checking Heisenberg commutation relations. • The AGT correspondence was originally formulated in terms of Nekrasov partition functions. Their mathematical background is given in [51]. • In view of the geometric Satake correspondence for affine Kac-Moody groups [10], the equivariant intersection cohomology group IH∗G×C× ×C× of instanton moduli spaces of R4 /Z should be understood in terms of representations of the affine Lie algebra of g and the corresponding generalized W-algebra. We believe that the necessary technical tools are more or less established in [12], but they still need to be worked out in detail. Anyhow, one certainly needs knowledge of W-algebras in order to study their generalizations.

References [1] L. F. Alday, D. Gaiotto, and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010), no. 2, 167–197. ←384, 385 [2] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. ←389 [3] S. Bando, Einstein-Hermitian metrics on noncompact Kähler manifolds, Einstein metrics and YangMills connections (Sanda, 1990), 1993, pp. 27–33. ←388 [4] V. Baranovsky, Moduli of sheaves on surfaces and action of the oscillator algebra, J. Differential Geom. 55 (2000), no. 2, 193–227. ←383, 392, 393, 394 [5] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. ←383 [6] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480– 497. ←399 [7] T. Braden, A. Licata, N. Proudfoot, and B. Webster, Quantizations of conical symplectic resolutions II: category O and symplectic duality (with an appendix by I. Losev), Astérisque 384 (2016), 75–179, available at arXiv:1407.0964. ←387, 413 [8] T. Braden and C. Mautner, Ringel duality for perverse sheaves on hypertoric varieties. in preparation. ←414

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References

435

[9] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209– 216. ←385, 406, 408, 409 [10] A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian I: transversal slices via instantons on Ak -singularities, Duke Math. J. 152 (2010), no. 2, 175–206. ←384, 385, 413, 425, 426, 427, 434 [11] A. Braverman, M. Finkelberg, and D. Gaitsgory, Uhlenbeck spaces via affine Lie algebras, The unity of mathematics, 2006, pp. 17–135. see arXiv:math/0301176 for erratum. ←387, 389, 424 [12] A. Braverman, M. Finkelberg, and H. Nakajima, Instanton moduli spaces and W-algebras, Astérisque 385 (2016), vii + 128, available at arXiv:1406.2381. ←382, 386, 387, 405, 411, 412, 420, 421, 422, 423, 424, 428, 429, 430, 431, 434 [13] A. Braverman, M. Finkelberg, and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II, Astérisque (January 2016), available at arXiv:1601.03586. ←386 [14] E. Carlsson and A. Okounkov, Exts and vertex operators, Duke Math. J. 161 (2012), no. 9, 1797–1815. ←433, 434 [15] J. B. Carrell and A. J. Sommese, C∗ -actions, Math. Scand. 43 (1978/79), no. 1, 49–59. ←399 [16] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhäuser Boston Inc., Boston, MA, 1997. ←383, 397, 401, 404, 407 [17] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. ←382 [18] V. Drinfeld and D. Gaitsgory, On a theorem of Braden, ArXiv e-prints (August 2013), available at arXiv:1308.3786. ←409 [19] V. G. Drinfel  d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064. ←385 [20] B. L. Feigin and A. I. Tsymbaliuk, Equivariant K-theory of Hilbert schemes via shuffle algebra, Kyoto J. Math. 51 (2011), no. 4, 831–854. ←385 [21] E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves, Second, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. ←383, 386, 428, 429, 431, 434 [22] B. Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), no. 1, 167–186. ←389 [23] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253–280. ←406 [24] L. Göttsche, H. Nakajima, and K. Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), no. 3, 343–390. ←384 [25] L. Göttsche, H. Nakajima, and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 307–359. ←384 [26] I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275–291. ←383, 393, 394 [27] V. G. Kac, Infinite-dimensional Lie algebras, Third, Cambridge University Press, Cambridge, 1990. ←426, 427 [28] D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 600 (2006), 135–156. ←392, 401 [29] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. ←406, 407 [30] G. Laumon, Comparaison de caractéristiques d’Euler-Poincaré en cohomologie l-adique, C.R.Acad. Sci. Paris Sér. I Math 292 (1981), no. 3, 209–212. ←424, 427 [31] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), no. 1, 157–207. ←418 [32] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. ←394 [33] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. ←410 [34] G. Lusztig, Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193–237. ←410 [35] G. Lusztig, Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994), 1995, pp. 217–275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 145–202; MR0972345 (90e:22029)]. ←383

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436

References

[36] D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, ArXiv e-prints (November 2012), available at arXiv:1211.1287. ←382, 385, 395, 396, 397, 415, 417, 418, 419, 420 [37] M. B. McBreen, Quantum Cohomology of Hypertoric Varieties and Geometric Representations of Yangians, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Princeton University. ←415, 420 [38] I. Mirkovi´c and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. ←408, 411, 412, 413, 414 [39] A. Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 143, American Mathematical Society, Providence, RI, 2007. ←415 [40] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. ←383, 394 [41] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), no. 2, 379–388. ←383, 393, 394 [42] H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560.←414 [43] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. ←383, 391, 392, 394, 401, 418 [44] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238 (electronic). ←383 [45] H. Nakajima, Quiver varieties and tensor products, Invent. Math. 146 (2001), no. 2, 399–449. ←399 [46] H. Nakajima, Quiver varieties and branching, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 003, 37. ←384, 413 [47] H. Nakajima, Quiver varieties and tensor products, II, Symmetries, integrable systems and representations, 2013, pp. 403–428. ←382, 385, 407 [48] H. Nakajima, More lectures on Hilbert schemes of points on surfaces, ArXiv e-prints (January 2014), available at arXiv:1401.6782. ←383, 418 [49] H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (March 2016), no. 3, 595–669, available at arXiv:1503.03676. ←386, 413, 414 [50] H. Nakajima and Y. Takayama, Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A, ArXiv e-prints (June 2016), available at arXiv:1606.02002. ←387 [51] H. Nakajima and K. Yoshioka, Lectures on instanton counting, Algebraic structures and moduli spaces, 2004, pp. 31–101. ←434 [52] Y. Namikawa, Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), no. 1, 51–85. ←403 [53] A. Negu¸t, Exts and the AGT Relations, Letters in Mathematical Physics (June 2016), available at arXiv:1510.05482. ←433, 434 [54] N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5, 831–864. ←383 [55] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. ←394 [56] O. Schiffmann and E. Vasserot, Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2 , Publ. Math. Inst. Hautes Études Sci. 118 (2013), 213–342. ←385, 395, 396 [57] O. Schiffmann and E. Vasserot, The elliptic Hall algebra and the K-theory of the Hilbert scheme of A2 , Duke Math. J. 162 (2013), no. 2, 279–366. ←385 [58] W. Slofstra, A Brylinski filtration for affine Kac-Moody algebras, Adv. Math. 229 (2012), no. 2, 968–983. ←384, 385 [59] Y. Tachikawa, A pseudo-mathematical pseudo-review on 4d N = 2 supersymmetric quantum field theories. available at http://member.ipmu.jp/yuji.tachikawa/not-on-arxiv.html. ←431, 432 [60] J. Teschner (ed.), New dualities of supersymmetric gauge theories, Mathematical Physics Studies, Springer, Cham, 2016. ←431, 434 [61] C. Vafa and E. Witten, A strong coupling test of S-duality, Nuclear Phys. B 431 (1994), no. 1-2, 3–77. ←433 [62] M. Varagnolo and E. Vasserot, Perverse sheaves and quantum Grothendieck rings, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), 2003, pp. 345–365. ←410 [63] M. Varagnolo, Quiver varieties and Yangians, Lett. Math. Phys. 53 (2000), no. 4, 273–283. ←383 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected]

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This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.

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