Mathematical Surveys

and

Volume 228

Hilbert Schemes of

Dimensional Lie Algebras

Zhenbo Qin

Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

10.1090/surv/228

Mathematical Surveys

and Monographs

Volume 228

Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

Zhenbo Qin

EDITORIAL COMMITTEE

Robert GuralnickMichael A. Singer, Chair

Benjamin SudakovConstantin Teleman

Michael I. Weinstein

The author was supported in part by a Collaboration Grant for Mathematicians (AwardNumber: 268702) from the Simons Foundation and by a Research Council Grant (AwardNumber: URC-17-084-n) from the University of Missouri.

2010 Mathematics Subject Classification. Primary 14C05, 17B65;Secondary 14F43, 14J60, 14N35, 17B69.

For additional information and updates on this book, visitwww.ams.org/bookpages/surv-228

Library of Congress Cataloging-in-Publication Data

Names: Qin, Zhenbo, author.Title: Hilbert schemes of points and infinite dimensional lie algebras / Zhenbo Qin.Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Mathe-

matical surveys and monographs; volume 228 | Includes bibliographical references and index.Identifiers: LCCN 2017036491 | ISBN 9781470441883 (alk. paper)Subjects: LCSH: Hilbert schemes. | Schemes (Algebraic geometry) | Lie algebras. | AMS: Alge-

braic geometry – Cycles and subschemes – Parametrization (Chow and Hilbert schemes). msc| Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Infinite-dimensionalLie (super)algebras. msc | Algebraic geometry – (Co)homology theory – Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies).msc | Algebraic geometry – Surfaces and higher-dimensional varieties – Vector bundles on sur-faces and higher-dimensional varieties, and their moduli. msc | Algebraic geometry – Projectiveand enumerative geometry – Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants. msc | Nonassociative rings and algebras –Lie algebras and Lie superalgebras – Vertex operators; vertex operator algebras and relatedstructures. msc

Classification: LCC QA564 .Q56 2018 | DDC 516.3/5–dc23LC record available at https://lccn.loc.gov/2017036491

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Contents

Preface ix

Part 1. Hilbert schemes of points on surfaces 1

Chapter 1. Basic results on Hilbert schemes of points 31.1. Partitions 31.2. The ring of symmetric functions 51.3. Symmetric products 71.4. Hilbert schemes of points 101.5. Incidence Hilbert schemes 17

Chapter 2. The nef cone and flip structure of (P2)[n] 192.1. Curves homologous to βn 192.2. The nef cone of (P2)[n] 282.3. Curves homologous to β� − (n− 1)βn 322.4. A flip structure on (P2)[n] when n ≥ 3 35

Part 2. Hilbert schemes and infinite dimensional Lie algebras 41

Chapter 3. Hilbert schemes and infinite dimensional Lie algebras 433.1. Affine Lie algebra action of Nakajima 433.2. Heisenberg algebras of Nakajima and Grojnowski 463.3. Geometric interpretations of Heisenberg monomial classes 553.4. The homology classes of curves in Hilbert schemes 583.5. Virasoro algebras of Lehn 613.6. Higher order derivatives of Heisenberg operators 633.7. The Ext vertex operators of Carlsson and Okounkov 69

Chapter 4. Chern character operators 734.1. Chern character operators 734.2. Chern characters 804.3. Characteristic classes of tautological bundles 874.4. W algebras and Hilbert schemes 91

Chapter 5. Multiple q-zeta values and Hilbert schemes 995.1. Okounkov’s conjecture 995.2. The series Fα1,...,αN

k1,...,kN(q) 102

5.3. The reduced series⟨chL1

k1· · · chLN

kN

⟩′118

Chapter 6. Lie algebras and incidence Hilbert schemes 1216.1. Heisenberg algebra actions for incidence Hilbert schemes 121

v

vi CONTENTS

6.2. A translation operator for incidence Hilbert schemes 1296.3. Lie algebras and incidence Hilbert schemes 137

Part 3. Cohomology rings of Hilbert schemes of points 139

Chapter 7. The cohomology rings of Hilbert schemes of pointson surfaces 141

7.1. Two sets of ring generators for the cohomology 1417.2. The Hilbert ring 1467.3. Approach of Lehn-Sorger via graded Frobenius algebras 1497.4. Approach of Costello-Grojnowski via Calogero-Sutherland

operators 154

Chapter 8. Ideals of the cohomology rings of Hilbert schemes 1578.1. The cohomology ring of the Hilbert scheme (C2)[n] 1578.2. Ideals in H∗(X [n]) for a projective surface X 1618.3. Relation with the cohomology ring of the Hilbert scheme (C2)[n] 1648.4. Partial n-independence of structure constants for X projective 1668.5. Applications to quasi-projective surfaces with the S-property 171

Chapter 9. Integral cohomology of Hilbert schemes 1759.1. Integral operators 1759.2. Integral operators involving only divisors in H2(X) 1809.3. Integrality of mλ,α for integral α 1849.4. Unimodularity 1859.5. Integral bases for the cohomology of Hilbert schemes 1909.6. Comparison of two integral bases of H∗((P2)[n];Z) 191

Chapter 10. The ring structure of H∗orb(X

(n)) 20310.1. Generalities 20310.2. The Heisenberg algebra 20510.3. The cohomology classes ηn(γ) and Ok(α, n) 20610.4. Interactions between Heisenberg algebra and Ok(γ) 20910.5. The ring structure of H∗

orb(X(n)) 212

10.6. The W algebras 214

Part 4. Equivariant cohomology of the Hilbert schemesof points 217

Chapter 11. Equivariant cohomology of Hilbert schemes 21911.1. Equivariant cohomology rings of Hilbert schemes 21911.2. Heisenberg algebras in equivariant setting 22411.3. Equivariant cohomology and Jack polynomials 225

Chapter 12. Hilbert/Gromov-Witten correspondence 23112.1. A brief introduction to Gromov-Witten theory 23212.2. The Hilbert/Gromov-Witten correspondence 23312.3. The N -point functions and the multi-point trace functions 23812.4. Equivariant intersection and τ -functions of 2-Toda hierarchies 24112.5. Numerical aspects of Hilbert/Gromov-Witten correspondence 244

CONTENTS vii

12.6. Relation to the Hurwitz numbers of P1 247

Part 5. Gromov-Witten theory of the Hilbert schemesof points 251

Chapter 13. Cosection localization for the Hilbert schemes of points 25313.1. Cosection localization of Kiem and J. Li 25313.2. Vanishing of Gromov-Witten invariants when pg(X) > 0 25713.3. Intersections on some moduli space of genus-1 stable maps 26113.4. Gromov-Witten invariants of the Hilbert scheme X [2] 265

Chapter 14. Equivariant quantum operator of Okounkov-Pandharipande 27114.1. Equivariant quantum cohomology of the Hilbert scheme (C2)[n] 27114.2. Equivalence of four theories 27514.3. The quantum differential equation of Hilbert schemes of points 278

Chapter 15. The genus-0 extremal Gromov-Witten invariants 28315.1. 1-point genus-0 extremal Gromov-Witten invariants 28315.2. 2-point genus-0 extremal invariants of J. Li and W.-P. Li 29415.3. The structure of the genus-0 extremal Gromov-Witten invariants 301

Chapter 16. Ruan’s Cohomological Crepant Resolution Conjecture 30716.1. The quantum corrected cohomology ring H∗

ρn(X [n]) 308

16.2. The commutator [Gk(α), a−1(β)] 31016.3. Ruan’s Cohomological Crepant Resolution Conjecture 322

Bibliography 325

Index 335

Preface

Since the pioneering work of Grothendieck [Grot], Hilbert schemes, which pa-rametrize subschemes in algebraic varieties, have been studied extensively as evi-denced in the survey [Iar2]. Viewed also as the simplest moduli spaces of sheaves,they are fundamental objects in algebraic geometry, play essential roles in variousimportant enumerative problems, and provide testing grounds for many interestingconjectures in algebraic geometry and its interplay with theoretical physics. Apartfrom the Hilbert schemes of curves on 3-folds over which the Donaldson-Thomastheory is defined and investigated [MNOP1,MNOP2], much attention has beenconcentrated on Hilbert schemes of points, i.e., those Hilbert schemes parametriz-ing 0-dimensional closed subschemes. When the dimension of the variety is at leastthree, the corresponding Hilbert schemes of points are in general singular. Whenthe variety is a smooth curve, the corresponding Hilbert schemes of points coincidewith the symmetric products of the curve. When X is a smooth algebraic surface,the Hilbert scheme X [n] parametrizing length-n 0-dimensional closed subschemesof X is irreducible and smooth [Bri,Fog1, Iar1]. Moreover, the Hilbert-Chowmorphism ρn from X [n] to the symmetric product X(n), which sends an elementin X [n] to its support (counted with multiplicities) in X(n), is a crepant desin-gularization of X(n). Many fundamental aspects of X [n] such as its cohomologygroups, Chow groups, motive, cobordism class, and relations with algebraic com-binatorics, the McKay correspondence and integrable systems have been analyzed[CoG,dCM,EGL,ES1,Got1,Hai1,Hai2,Mar1].

In the seminal papers [Groj,Nak3] which were motivated by [Nak1,Nak2,Nak4] regarding the construction of representations of affine Lie algebras on thehomology groups of the moduli spaces of instantons on ALE spaces (equivalently,on the homology groups of quiver varieties), Grojnowski and Nakajima geomet-rically constructed Heisenberg algebra actions on the cohomology of the Hilbertschemes X [n], where X denotes a smooth algebraic surface. Their geometric con-structions started a whole new chapter investigating interplays between the Hilbertschemes of points and infinite dimensional Lie algebras. Subsequently, using theHilbert schemes X [n], Lehn [Leh1] geometrically constructed the Virasoro algebrasand the boundary operator, W.-P. Li, W. Wang and the author [LQW1,LQW4]constructed the Chern character operators and the W algebras, and Carlsson andOkounkov [Car1,Car2,CO] constructed the Ext vertex operators. As noted in[FW, Introduction] and [Mat, Section 5], Lehn’s boundary operator is a versionof the bosonized Calogero-Sutherland operator. The Chern character operators arevertex operators of higher conformal weights, and the W algebras are higher-spingeneralizations of Lehn’s Virasoro algebras. The Ext vertex operators of Carlssonand Okounkov are motivated by the study of Nekrasov partition functions (which

ix

x PREFACE

arise from supersymmetric quantum gauge theory) in the setting of Hilbert schemesof points.

All of these operators and algebras are powerful tools in understanding thefiner geometric properties of the Hilbert schemes X [n]. Indeed, the Heisenbergalgebras serve as a fundamental language in describing the homology and cohomol-ogy classes of X [n], which is crucial in studying the Gromov-Witten theory of X [n]

[Cheo2,ELQ,HLQ,LL,LQ1,LQ5,OP3,OP4]. The Virasoro operators and theChern character operators determine the cup products and ring structures of thecohomology of X [n] [CoG,LQW1,LQW2,LQW3,LQW5,LS1,LS2]. The Extvertex operators provide a very useful approach in understanding the intersectiontheory of X [n] when the tangent bundle of X [n] is involved [Car1,Car2,QY].

Two excellent books on Hilbert schemes of points exist. The first one isGottsche’s book [Got2] which includes fundamental facts about X [n], the Bettinumbers of X [n] when X is a surface, and the computation of the homology andChow rings of Hilbert schemes. The second is Nakajima’s book [Nak5] whichcontains his Heisenberg algebra constructions, symmetric products (in the Hilbertscheme X [n]) of an embedded curve in a surface X, and interactions with singu-larities, symplectic geometry and the ring of symmetric functions. We also refer to[EG,Leh2] for surveys on Hilbert schemes of points, the Heisenberg algebras andthe Virasoro algebras, and to [Nak7] for a survey on the equivariant cohomologyof the Hilbert schemes, the Heisenberg algebras, the Virasoro algebras and someinteresting applications to algebraic combinatorics.

The purpose of this book is to present a detailed survey of the developments,in the context of interactions between Hilbert schemes of points and infinite dimen-sional Lie algebras, that appeared after the book [Nak5]. This book contains 5parts consisting of 16 chapters. Part 1 deals with the basics of the Hilbert schemesof points and some geometry of the Hilbert schemes of points on the projectiveplane. Part 2 is devoted to the constructions of various infinite dimensional Lie al-gebra actions on the cohomology of the Hilbert schemes of points on surfaces, andto the connections with multiple q-zeta values. It includes Nakajima’s affine Liealgebra actions on the homology of quiver varieties as a motivation and backgroundmaterial, the Heisenberg algebras of Grojnowski and Nakajima, the boundary op-erator and the Virasoro algebras of Lehn, the Ext vertex operators of Carlsson andOkounkov, and the Chern character operators and the W algebras of W.-P. Li,W. Wang and the author. Part 3 studies the cohomology ring structure of theseHilbert schemes, such as the ring generators and the ideals, the approach of Lehnand Sorger when the canonical divisor of the surface is numerically trivial, the ap-proach of Costello and Grojnowski in terms of the Calogero-Sutherland operatorsand the Dunkl-Cherednik operators, the integral basis for the cohomology group,and the orbifold cohomology ring of the symmetric product of a surface. Part 4is about the equivariant cohomology of the Hilbert schemes via Jack polynomials,the Hilbert/Gromov-Witten correspondence between the equivariant cohomologyof the Hilbert schemes of points on the affine plane and the Gromov-Witten theoryof curves, and applications to the Hurwitz numbers of the projective line. Part 5includes the cosection localization technique of Kiem and J. Li, the Gromov-Witten

PREFACE xi

theory of the Hilbert schemes of points, the equivariant quantum corrected bound-ary operator and the quantum differential equations of Okounkov and Pandhari-pande, the quantum corrected boundary operator of J. Li and W.-P. Li, and theproof of Ruan’s Cohomological Crepant Resolution Conjecture.

This book is suitable for graduate students and researchers in algebraic ge-ometry, representation theory, algebraic combinatorics, topology, number theoryand theoretical physics. Furthermore, a semester of an advanced course on Hilbertschemes of points and infinite dimensional Lie algebras may be organized from se-lected chapters in this book, e.g., Chapter 1, Chapter 3, Chapter 4, Chapter 7,Chapter 8, Chapter 11 and Chapter 12.

The author would like to thank all of his collaborators on works related tothe Hilbert schemes of points, without whom this book would be impossible: DanEdidin, Jianxun Hu, Wei-Ping Li, Yuping Tu, Weiqiang Wang, Fei Yu and QiZhang. In addition, thanks are due to Lie Fu, Wei-Ping Li, Hiraku Nakajima, BoiarQin, Wenzer Qin, Weiqiang Wang and the two anonymous referees for carefullyreading the manuscript and providing useful comments which have greatly improvedthe exposition of the book.

The author also thanks Sergei Gelfand, Christine M. Thivierge and the EditorialCommittee of the American Mathematical Society for their editorial guidance.

December 14, 2017

Zhenbo Qin

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Index

N-point function, 238

W algebra, 92τ -function, 242

k-very ample, 16q-trace, 237

ADHM datum, 44adjoint operator, 46

affine Kac-Moody algebra ˜sl�, 45alternating character, 160

arm colength, 4

bi-degree, 46

Borel-Moore homology, 157boson-fermion correspondence, 236

boundary of Hilbert schemes, 15boundary operator, 61

Calogero-Sutherland operator, 155Cartan matrix, 44

Chern character operator, 73Chevalley generators, 45

class algebra, 160Cohomological Crepant Resolution

Conjecture, 322

content, 4convolution product, 160

Crepant Resolution Conjecture, 307

curvi-linear ideals, 12

degeneracy locus, 254, 257degree defect, 152

degree of a permutation, 152degree shift number, 204

Divisor Axiom, 233

dominance ordering, 3

elementary symmetric function, 5equivariant cohomology, 220

equivariant quantum corrected boundaryoperator, 273

Euler class, 151Euler-Poincare characteristic, 151

evaluation map, 232excess dimension, 233

expected dimension, 232Ext operator, 70

FH ring, 173forgetful map, 233forgotten symmetric function, 7Fundamental Class Axiom, 232

graded Frobenius algebra, 150, 322graph defect, 152

Gromov-Witten invariants, 232

Hecke correspondence, 45Heisenberg monomial classes, 48Heisenberg operators, 47Hilbert ring, 147Hilbert scheme of α-points, 301

Hilbert scheme of Λ-points, 301Hilbert scheme of n points, 10Hilbert scheme, punctual, 12Hilbert-Chow morphism, 12Hilbert/Gromov-Witten correspondence,

237, 276Hodge bundle, 261hook length, 4Hurwitz cover, 247

Hurwitz number, 248

incidence Hilbert scheme, 17induction map, 204integral basis, 175integral class, 175integral operator, 175

Jack symmetric function, 7Jucys-Murphy elements, 207

leg colength, 4leg length, 4Lie superalgebra bracket, 46local Gromov-Witten theory, 276

monomial symmetric function, 6multi-point trace function, 240multiple q-zeta value, 99multiple zeta value, 99

335

336 INDEX

Mumford principle, 141

nef, 16

nef and big, 16normally ordered product, 61, 93

operator product expansion, 96orbifold, 203orbifold cohomology group, 204orbifold cohomology ring, 204

partition, 3partition, d-dimensional, 3partition, generalized, 4Poincare polynomial, 7power symmetric function, 6

quantum corrected boundary operator, 296quantum corrected cohomology ring, 308quantum corrected product, 308quantum differential equation, 278quantum-mechanical Calogero-Sutherland

operator, 278quiver, 43quiver variety, 44

Riemann zeta function, 99ring of symmetric functions, 5

S-property, 171Schur function, 6stable map, 232standard decomposition, 256symmetric product, 7

tautological bundle, 15transfer properties, 77translation operator, 130

universal linear combination, 76, 162

vacuum expectation, 236Virasoro operator, 61virtual fundamental class, 232

weight, 100

Young diagram, 4

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For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/survseries/.

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Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes X [n ] of collections of n points (zero-dimensional subschemes) in a smooth algebraic surface X . Schemes X [n ] turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others.

This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of X [n ] , including the construction of the action of infinite dimen-sional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of X [n ] and the Gromov–Witten correspondence. The last part of the book presents results about quantum cohomology of X [n ] and related questions.

The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.

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