Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 10
Part  1 .  Hilbert schemes of points on surfaces......Page 14
1.1. Partitions......Page 16
1.2. The ring of symmetric functions......Page 18
1.3. Symmetric products......Page 20
1.4. Hilbert schemes of points......Page 23
1.5. Incidence Hilbert schemes......Page 30
2.1. Curves homologous to _{}......Page 32
2.2. The nef cone of (ℙ²)^{[]}......Page 41
2.3. Curves homologous to _{ℓ}-(-1)_{}......Page 45
2.4. A flip structure on (ℙ²)^{[]} when ≥3......Page 48
Part  2 .  Hilbert schemes and infinite dimensional Lie algebras......Page 54
3.1. Affine Lie algebra action of Nakajima......Page 56
3.2. Heisenberg algebras of Nakajima and Grojnowski......Page 59
3.3. Geometric interpretations of Heisenberg monomial classes......Page 68
3.4. The homology classes of curves in Hilbert schemes......Page 71
3.5. Virasoro algebras of Lehn......Page 74
3.6. Higher order derivatives of Heisenberg operators......Page 76
3.7. The Ext vertex operators of Carlsson and Okounkov......Page 82
4.1. Chern character operators......Page 86
4.2. Chern characters......Page 93
4.3. Characteristic classes of tautological bundles......Page 100
4.4.  algebras and Hilbert schemes......Page 104
5.1. Okounkov’s conjecture......Page 112
5.2. The series ^{₁,…,_{}}_{₁,…,_{}}()......Page 115
5.3. The reduced series \big⟨ℎ_{₁}^{₁}\cdotsℎ_{_{}}^{_{}}\big⟩’......Page 131
6.1. Heisenberg algebra actions for incidence Hilbert schemes......Page 134
6.2. A translation operator for incidence Hilbert schemes......Page 142
6.3. Lie algebras and incidence Hilbert schemes......Page 150
Part  3 .  Cohomology rings of Hilbert schemes of points......Page 152
7.1. Two sets of ring generators for the cohomology......Page 154
7.2. The Hilbert ring......Page 159
7.3. Approach of Lehn-Sorger via graded Frobenius algebras......Page 162
7.4. Approach of Costello-Grojnowski via Calogero-Sutherland operators......Page 167
8.1. The cohomology ring of the Hilbert scheme (ℂ²)^{[]}......Page 170
8.2. Ideals in *(^{[]}) for a projective surface ......Page 174
8.3. Relation with the cohomology ring of the Hilbert scheme (ℂ²)^{[]}......Page 177
8.4. Partial -independence of structure constants for  projective......Page 179
8.5. Applications to quasi-projective surfaces with the S-property......Page 184
9.1. Integral operators......Page 188
9.2. Integral operators involving only divisors in ²()......Page 193
9.3. Integrality of _{,} for integral ......Page 197
9.4. Unimodularity......Page 198
9.5. Integral bases for the cohomology of Hilbert schemes......Page 203
9.6. Comparison of two integral bases of *((ℙ²)^{[]};ℤ)......Page 204
10.1. Generalities......Page 216
10.2. The Heisenberg algebra......Page 218
10.3. The cohomology classes _{}() and _{}(,)......Page 219
10.4. Interactions between Heisenberg algebra and _{}()......Page 222
10.5. The ring structure of *_{}(⁽ⁿ⁾)......Page 225
10.6. The  algebras......Page 227
Part  4 .  Equivariant cohomology of the Hilbert schemes of points......Page 230
11.1. Equivariant cohomology rings of Hilbert schemes......Page 232
11.2. Heisenberg algebras in equivariant setting......Page 237
11.3. Equivariant cohomology and Jack polynomials......Page 238
Chapter 12. Hilbert/Gromov-Witten correspondence......Page 244
12.1. A brief introduction to Gromov-Witten theory......Page 245
12.2. The Hilbert/Gromov-Witten correspondence......Page 246
12.3. The -point functions and the multi-point trace functions......Page 251
12.4. Equivariant intersection and -functions of 2-Toda hierarchies......Page 254
12.5. Numerical aspects of Hilbert/Gromov-Witten correspondence......Page 257
12.6. Relation to the Hurwitz numbers of ℙ¹......Page 260
Part  5 .  Gromov-Witten theory of the Hilbert schemes of points......Page 264
13.1. Cosection localization of Kiem and J. Li......Page 266
13.2. Vanishing of Gromov-Witten invariants when _{}()>0......Page 270
13.3. Intersections on some moduli space of genus-1 stable maps......Page 274
13.4. Gromov-Witten invariants of the Hilbert scheme ^{[2]}......Page 278
14.1. Equivariant quantum cohomology of the Hilbert scheme (ℂ²)^{[]}......Page 284
14.2. Equivalence of four theories......Page 288
14.3. The quantum differential equation of Hilbert schemes of points......Page 291
15.1. 1-point genus-0 extremal Gromov-Witten invariants......Page 296
15.2. 2-point genus-0 extremal invariants of J. Li and W.-P. Li......Page 307
15.3. The structure of the genus-0 extremal Gromov-Witten invariants......Page 314
Chapter 16. Ruan’s Cohomological Crepant Resolution Conjecture......Page 320
16.1. The quantum corrected cohomology ring *_{_{}}(^{[]})......Page 321
16.2. The commutator [_{}(),₋₁()]......Page 323
16.3. Ruan’s Cohomological Crepant Resolution Conjecture......Page 335
Bibliography......Page 338
Index......Page 348
Back Cover......Page 351