طرح‌های هیلبرت نقاط و جبرهای لی با بعد نامتناهی

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. Cover 1 Title page 4 Contents 6 Preface 10 Part 1 . Hilbert schemes of points on surfaces 14 Chapter 1. Basic results on Hilbert schemes of points 16 1.1. Partitions 16 1.2. The ring of symmetric functions 18 1.3. Symmetric products 20 1.4. Hilbert schemes of points 23 1.5. Incidence Hilbert schemes 30 Chapter 2. The nef cone and flip structure of (P2)^{[n]} 32 2.1. Curves homologous to β_{n} 32 2.2. The nef cone of (P2)^{[n]} 41 2.3. Curves homologous to β_{l}-(n-1)β_{n} 45 2.4. A flip structure on (P2)^{[n]} when n≥3 48 Part 2 . Hilbert schemes and infinite dimensional Lie algebras 54 Chapter 3. Hilbert schemes and infinite dimensional Lie algebras 56 3.1. Affine Lie algebra action of Nakajima 56 3.2. Heisenberg algebras of Nakajima and Grojnowski 59 3.3. Geometric interpretations of Heisenberg monomial classes 68 3.4. The homology classes of curves in Hilbert schemes 71 3.5. Virasoro algebras of Lehn 74 3.6. Higher order derivatives of Heisenberg operators 76 3.7. The Ext vertex operators of Carlsson and Okounkov 82 Chapter 4. Chern character operators 86 4.1. Chern character operators 86 4.2. Chern characters 93 4.3. Characteristic classes of tautological bundles 100 4.4. W algebras and Hilbert schemes 104 Chapter 5. Multiple q-zeta values and Hilbert schemes 112 5.1. Okounkov’s conjecture 112 5.2. The series F^{α1,...,α_{N}}_{k1,...,k_{N}}(q) 115 5.3. The reduced series \big⟨ch_{k1}^{L1}\cdotsch_{k_{N}}^{L_{N}}\big⟩’ 131 Chapter 6. Lie algebras and incidence Hilbert schemes 134 6.1. Heisenberg algebra actions for incidence Hilbert schemes 134 6.2. A translation operator for incidence Hilbert schemes 142 6.3. Lie algebras and incidence Hilbert schemes 150 Part 3 . Cohomology rings of Hilbert schemes of points 152 Chapter 7. The cohomology rings of Hilbert schemes of points on surfaces 154 7.1. Two sets of ring generators for the cohomology 154 7.2. The Hilbert ring 159 7.3. Approach of Lehn-Sorger via graded Frobenius algebras 162 7.4. Approach of Costello-Grojnowski via Calogero-Sutherland operators 167 Chapter 8. Ideals of the cohomology rings of Hilbert schemes 170 8.1. The cohomology ring of the Hilbert scheme (C2)^{[n]} 170 8.2. Ideals in H*(X^{[n]}) for a projective surface X 174 8.3. Relation with the cohomology ring of the Hilbert scheme (C2)^{[n]} 177 8.4. Partial n-independence of structure constants for X projective 179 8.5. Applications to quasi-projective surfaces with the S-property 184 Chapter 9. Integral cohomology of Hilbert schemes 188 9.1. Integral operators 188 9.2. Integral operators involving only divisors in H2(X) 193 9.3. Integrality of m_{λ,α} for integral α 197 9.4. Unimodularity 198 9.5. Integral bases for the cohomology of Hilbert schemes 203 9.6. Comparison of two integral bases of H*((P2)^{[n]};Z) 204 Chapter 10. The ring structure of H*_{orb}(X(n)) 216 10.1. Generalities 216 10.2. The Heisenberg algebra 218 10.3. The cohomology classes η_{n}(γ) and O_{k}(α,n) 219 10.4. Interactions between Heisenberg algebra and O_{k}(γ) 222 10.5. The ring structure of H*_{orb}(X(n)) 225 10.6. The W algebras 227 Part 4 . Equivariant cohomology of the Hilbert schemes of points 230 Chapter 11. Equivariant cohomology of Hilbert schemes 232 11.1. Equivariant cohomology rings of Hilbert schemes 232 11.2. Heisenberg algebras in equivariant setting 237 11.3. Equivariant cohomology and Jack polynomials 238 Chapter 12. Hilbert/Gromov-Witten correspondence 244 12.1. A brief introduction to Gromov-Witten theory 245 12.2. The Hilbert/Gromov-Witten correspondence 246 12.3. The N-point functions and the multi-point trace functions 251 12.4. Equivariant intersection and τ-functions of 2-Toda hierarchies 254 12.5. Numerical aspects of Hilbert/Gromov-Witten correspondence 257 12.6. Relation to the Hurwitz numbers of P1 260 Part 5 . Gromov-Witten theory of the Hilbert schemes of points 264 Chapter 13. Cosection localization for the Hilbert schemes of points 266 13.1. Cosection localization of Kiem and J. Li 266 13.2. Vanishing of Gromov-Witten invariants when p_{g}(X)>0 270 13.3. Intersections on some moduli space of genus-1 stable maps 274 13.4. Gromov-Witten invariants of the Hilbert scheme X^{[2]} 278 Chapter 14. Equivariant quantum operator of Okounkov-Pandharipande 284 14.1. Equivariant quantum cohomology of the Hilbert scheme (C2)^{[n]} 284 14.2. Equivalence of four theories 288 14.3. The quantum differential equation of Hilbert schemes of points 291 Chapter 15. The genus-0 extremal Gromov-Witten invariants 296 15.1. 1-point genus-0 extremal Gromov-Witten invariants 296 15.2. 2-point genus-0 extremal invariants of J. Li and W.-P. Li 307 15.3. The structure of the genus-0 extremal Gromov-Witten invariants 314 Chapter 16. Ruan’s Cohomological Crepant Resolution Conjecture 320 16.1. The quantum corrected cohomology ring H*_{ρ_{n}}(X^{[n]}) 321 16.2. The commutator [G_{k}(α),a−1(β)] 323 16.3. Ruan’s Cohomological Crepant Resolution Conjecture 335 Bibliography 338 Index 348 Back Cover 351 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