**Mathematical Surveys and Monographs**

Volume: 228;
2018;
336 pp;
Hardcover

MSC: Primary 14; 17;

Print ISBN: 978-1-4704-4188-3

Product Code: SURV/228

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4389-4
Product Code: SURV/228.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

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*Zhenbo Qin*

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 dimensional 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.

#### Readership

Graduate students and researchers interested in algebraic geometry.

#### Table of Contents

# Table of Contents

## Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Part 1 . Hilbert schemes of points on surfaces 114
- Part 2 . Hilbert schemes and infinite dimensional Lie algebras 4154
- Chapter 3. Hilbert schemes and infinite dimensional Lie algebras 4356
- 3.1. Affine Lie algebra action of Nakajima 4356
- 3.2. Heisenberg algebras of Nakajima and Grojnowski 4659
- 3.3. Geometric interpretations of Heisenberg monomial classes 5568
- 3.4. The homology classes of curves in Hilbert schemes 5871
- 3.5. Virasoro algebras of Lehn 6174
- 3.6. Higher order derivatives of Heisenberg operators 6376
- 3.7. The Ext vertex operators of Carlsson and Okounkov 6982

- Chapter 4. Chern character operators 7386
- Chapter 5. Multiple 𝑞-zeta values and Hilbert schemes 99112
- Chapter 6. Lie algebras and incidence Hilbert schemes 121134

- Part 3 . Cohomology rings of Hilbert schemes of points 139152
- Chapter 7. The cohomology rings of Hilbert schemes of points on surfaces 141154
- Chapter 8. Ideals of the cohomology rings of Hilbert schemes 157170
- 8.1. The cohomology ring of the Hilbert scheme (ℂ²)^{[𝕟]} 157170
- 8.2. Ideals in 𝐻*(𝑋^{[𝑛]}) for a projective surface 𝑋 161174
- 8.3. Relation with the cohomology ring of the Hilbert scheme (ℂ²)^{[𝕟]} 164177
- 8.4. Partial 𝑛-independence of structure constants for 𝑋 projective 166179
- 8.5. Applications to quasi-projective surfaces with the S-property 171184

- Chapter 9. Integral cohomology of Hilbert schemes 175188
- 9.1. Integral operators 175188
- 9.2. Integral operators involving only divisors in 𝐻²(𝑋) 180193
- 9.3. Integrality of 𝔪_{𝜆,𝛼} for integral 𝛼 184197
- 9.4. Unimodularity 185198
- 9.5. Integral bases for the cohomology of Hilbert schemes 190203
- 9.6. Comparison of two integral bases of 𝐻*((ℙ²)^{[𝕟]};ℤ) 191204

- Chapter 10. The ring structure of 𝐻*_{𝑜𝑟𝑏}(𝑋⁽ⁿ⁾) 203216

- Part 4 . Equivariant cohomology of the Hilbert schemes of points 217230
- Chapter 11. Equivariant cohomology of Hilbert schemes 219232
- Chapter 12. Hilbert/Gromov-Witten correspondence 231244
- 12.1. A brief introduction to Gromov-Witten theory 232245
- 12.2. The Hilbert/Gromov-Witten correspondence 233246
- 12.3. The 𝑁-point functions and the multi-point trace functions 238251
- 12.4. Equivariant intersection and 𝜏-functions of 2-Toda hierarchies 241254
- 12.5. Numerical aspects of Hilbert/Gromov-Witten correspondence 244257
- 12.6. Relation to the Hurwitz numbers of ℙ¹ 247260

- Part 5 . Gromov-Witten theory of the Hilbert schemes of points 251264
- Chapter 13. Cosection localization for the Hilbert schemes of points 253266
- Chapter 14. Equivariant quantum operator of Okounkov-Pandharipande 271284
- Chapter 15. The genus-0 extremal Gromov-Witten invariants 283296
- Chapter 16. Ruan’s Cohomological Crepant Resolution Conjecture 307320
- Bibliography 325338
- Index 335348

- Back Cover Back Cover1351