Part 3 Mathematical Physics,Algebraic Geometry and Other Topics 3
The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties&Bohan Fang,Chiu-Chu Melissa Liu, David Treumann and Eric Zaslow 3
1 Introduction 3
1.1 Outline 4
2 Mirror symmetry for toric manifolds 4
2.1 Hori-Vafa mirror 4
2.2 Categories in mirror symmetry 5
2.3 Results to date 8
3 T-duality 9
3.1 Moment polytope 9
3.2 Geometry of the open orbit 10
3.3 Statement of symplectic results 12
3.4 T-dual of an equivariant line bundle 14
4 Microlocalization 16
4.1 Algebraic preliminaries 16
4.2 The cast of categories 17
4.3 Fukaya-Oh theorem 19
4.4 Building the equivalence 20
4.5 Equivalence and the inverse functor 21
4.6 Singular support and characteristic cycles 22
4.7 Comments on technicalities 23
4.8 Statement of results 25
5 Coherent-constructible correspondence 25
6 Examples 28
6.1 Taking the mapping cone 28
6.2 Toric Fano surfaces 29
6.3 Hirzebruch surfaces 29
References 34
Superspace：a Comfortably Vast Algebraic Variety&T.Hübsch 39
1 Introduction 39
1.1 Basic ideas and definitions 40
1.2 The traditional superspace 42
2 Off-shell worldline supermultiplets 44
2.1 Adinkraic supermultiplets 45
2.2 Various hangings 46
2.3 Projected supermultiplets 48
2.4 Supermultiplets vs.superfields 49
3 Superspace,by construction 50
3.1 Superpartners of time 50
3.2 A telescoping deformation structure 55
3.3 Nontrivial superspace geometry 59
3.4 Higher-dimensional spacetime 62
4 The comfortably vast superspace 63
References 65
A Report on theYau-Zaslow Formula&Naichung Conan Leung 69
1 Yau-Zaslow formula and its generalizations 70
2 Yau-Zaslow approach 72
3 Matching method 72
4 Degeneration method 74
5 Calabi-Yau threefold method 77
6 Conclusions 78
References 79
Hermitian-Yang-Mills Connections on K？hler Manifolds&Jun Li 81
1 Introduction 81
1.1 Hermitian-Yang-Mills connections 81
1.2 HYM connections lead to stable bundles 83
1.3 Stable bundles and their moduli spaces 85
1.4 Flat bundles and stable bundles on curves 86
2 Donaldson-Uhlenbeck-Yau theorem 86
2.1 Donaldson's proof for algebraic surfaces 87
2.2 Uhlenbeck-Yau's proof for K？hler manifolds 88
3 Hermitian-Yang-Mills connections on curves 90
4 Hermitian-Yang-Mills connections on surfaces 92
4.1 Extending DUY correspondence 92
4.2 Stable topology of the moduli spaces 94
4.3 Donaldson polynomial invariants 95
5 HYM connections on high dimensional varieties 97
5.1 Extending the DUY correspondence in high dimensions 97
5.2 Donaldson-Thomas invariants 98
6 Concluding remark 99
References 99
Additivity and Relative Kodaira Dimensions&Tian-Jun Li and Weiyi Zhang 103
1 Introduction 103
2 Kodaira Dimensions and fiber bundles 104
2.1 kh for complex manifolds and Kt up to dimension 3 105
2.2 Ks for symplectic 4-manifolds 107
2.3 Additivity for a fiber bundle 109
3 Embedded symplectic surfaces and relative Kod.dim.in dim.4 112
3.1 Embedded symplectic surfaces and maximality 112
3.2 The adjoint class Kω+[F] 115
3.3 Existence and Uniqueness of relatively minimal model 122
3.4 ks(M,ω,F) 124
4 Relative Kod.dim.in dim.2 and fibrations over a surface 127
4.1 kt(F,D),Riemann-Hurwitz formula and Seifert fibrations 128
4.2 Lefschetz fibrations 130
References 133
Descendent Integrals and Tautological Rings of Moduli Spaces of Curves&Kefeng Liu and Hao Xu 137
1 Introduction 137
2 Intersection numbers and the Witten-Kontsevich theorem 138
2.1 Witten-Kontsevich theorem 139
2.2 Virasoro constraints 141
3 The n-point function 142
3.1 A recursive formula of n-point functions 142
3.2 An effective recursion formulae of descendent integrals 145
4 Hodge integrals 146
4.1 Faber's algorithm 146
4.2 Hodge integral formulae 147
5 Higher Weil-Petersson volumes 149
5.1 Generalization of Mirzakhani's recursion formula 149
5.2 Recursion formulae of higher Weil-Petersson volumes 151
6 Faber's conjecture on tautological rings 152
6.1 The Faber intersection number conjecture 153
6.2 Relations with n-point functions 154
7 Dimension of tautological rings 155
7.1 Ramanujan's mock theta functions 156
7.2 Asymptotics of tautological dimensions 158
8 Gromov-Witten invariants 161
8.1 Universal equations of Gromov-Witten invariants 162
8.2 Some vanishing identities 163
9 Witten's r-spin numbers 164
9.1 Generalized Witten's conjecture 165
9.2 An algorithm for computing Witten's r-spin numbers 166
Referenees 168
A General Voronoi Summation Formula for GL(n,？)&Stephen D.Miller and Wilfried Schmid 173
1 Introduction 173
2 Automorphic Distributions 178
3 Vanishing to infinite order 189
4 Classical proof of the formula 206
5 Adelic proof of the formula 211
References 223
Geometry of Holomorphic Isometries and Related Maps between Bounded Domains&Ngaiming Mok 225
1 Examples of holomorphic isometries 229
1.1 Examples of equivariant embeddings into the projective space 229
1.2 Non-standard holomorphic isometries of the Poincaré disk into polydisks 231
1.3 A non-standard holomorphic isometry of the Poincaré disk into a Siegel upper half-plane 232
1.4 Examples of holomorphic isometries with arbitrary normalizing constants λ>1 232
2 Analytic continuation of germs of holomorphic isometries 234
2.1 Analytic continuation of holomorphic isometries into the projective space equipped with the Fubini-Study metric 234
2.2 An extension and rigidity problem arising from commutators of modular correspondences 236
2.3 Analytic continuation of holomorphic isometries up to normalizing constants with respect to the Bergman metric-extension beyond the boundary 240
2.4 Canonically embeddable Bergman manifolds and Bergman meromorphic compactifications 246
3 Holomorphic isometries of the Poincaré disk into bounded symmetric domains 249
3.1 Structural equations on the norm of the second fundamental form and asymptotic vanishing order 249
3.2 Holomorphic isometries of the Poincaré disk into polydisks:structural results 250
3.3 Calculated examples on the norm of the second fundamental form 251
3.4 Holomorphic isometries of the Poincaré disk into polydisks:uniqueness results 253
3.5 Asymptotic total geodesy and applications 254
4 Measure-preserving algebraic correspondences on irreducible bounded symmetric domains 255
4.1 Statements of results 255
4.2 Extension results on strictly pseudoconvex algebraic hypersurfaces 256
4.3 Alexander-type extension results in the higher-rank case 257
4.4 Total geodesy of germs of measure-preserving holomorphic map from an irreducible bounded symmetric domain of dimension≥2 into its Cartesian products 259
5 Open problems 261
5.1 On the structure of the space of holomorphic isometries of the Poincaré disk into polydisks 261
5.2 On the second fundamental form and asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains 264
5.3 On germs of holomorphic maps preserving invariant(p,p)-forms 266
References 267
Abundance Conjecture&Yum-Tong Siu 271
0 Introduction 271
1 Curvature current and dichotomy 275
2 Gelfond-Schneider's technique of algebraic values of solutions of algebraically defined differential equations 279
3 Final step of the case of zero numerical Kodaira dimension 287
4 Numerically trivial foliations and fibrations for canonical line bundle 289
5 Curvature of zeroth direct image of relative canonical and pluricanonical bundle 291
6 Strict positivity of direct image of relative pluricanonical bundle along numerically trivial fibers in the base of numerically trivial fibration 302
7 Technique of Nevanlinna's first main theorem for proof of compactness of leaves of foliation 306
References 315
Sasaki-Einstein Geometry&James Sparks 319
1 Sasakian geometry 319
2 Constructions of Sasaki-Einstein manifolds 321
3 Obstructions 324
4 Sasaki-Einstein manifolds in string theory 326
References 327
A Simple Proof of the Chiral Gravity Conjecture&Andrew Strominger 329
Geometry of Grand Unification&Cumrun Vafa 335
1 Introduction 335
2 Standard model and gauge symmetry breaking 336
3 Flavors and hierarchy 337
4 Unification of gauge groups 338
5 String theory,forces,matter,and interactions 339
6 F-theory vacua 340
6.1 Matter fields 340
6.2 Yukawa couplings 342
7 Applications to particle physics 342
7.1 E-type singularity 343
7.2 Flavor hierarchy 343
7.3 Breaking to the standard model 344
8 Further issues 345
References 345
Quantum Invariance Under Flop Transitions&Chin-Lung Wang 347
1 Introduction 347
2 Ordinary flops:Genus zero theory 350
2.1 The canonical correspondence 350
2.2 The case of simple flops [9] 351
2.3 The topological defect 352
2.4 The extremal functions 353
2.5 Degeneration analysis 355
2.6 The local models 356
3 Calabi-Yau flops 358
3.1 The basic setup 358
3.2 I,P,J and their degrees 358
3.3 The CY condition and the mirror map 360
3.4 Example:Flops of type(P1,？(-7)),？(3)？(2)) 361
3.5 Proof of the main result in the example 365
References 369
The Problem Of Gauge Theory&Edward Witten 371
1 Yang-Mills equations 371
2 Classical phase space 373
3 Quantization 376
4 Nonperturbative approach 379
5 Breaking of conformal invariance and the mass gap 381
References 382
Part 4 Appendices 385
Shing-Tung Yau,a Manifold Man of Mathematics&Lizhen Ji and Kefeng Liu 385
1 Childhood and early school education 386
2 Middle school and college 387
3 Graduate school 390
4 Professional career 392
5 Major contributions to mathematics 395
6 Visits to China 404
7 Research centers and mathematics institutes 405
8 ICCM 408
9 Conferences and popular mathematics programs 409
10 Mathematics and Chinese literature 410
11 Family,friends and students 410
12 Summary 415
References 416
Perspectives on Geometric Analysis&Shing-Tung Yau 417
1 History and contributors of the subject 419
1.1 Founding fathers of the subject 419
1.2 Modern Contributors 421
2 Construction of functions in geometry 422
2.1 Polynomials from ambient space 423
2.2 Geometric construction of functions 426
2.3 Functions and tensors defined by linear differential equations 430
3 Mappings between manifolds and rigidity of geometric structures 446
3.1 Embedding 446
3.2 Rigidity of harmonic maps with negative curvature 449
3.3 Holomorphic maps 451
3.4 Harmonic maps from two dimensional surfaces and pseudoholomorphic curves 452
3.5 Morse theory for maps and topological applications 453
3.6 Wave maps 454
3.7 Integrable system 454
3.8 Regularity theory 455
4 Submanifolds defined by variational principles 455
4.1 Teichmüller space 455
4.2 Classical minimal surfaces in Euclidean space 456
4.3 Douglas-Morrey solution,embeddedness and application to topology of three manifolds 457
4.4 Surfaces related to classical relativity 458
4.5 Higher dimensional minimal subvarieties 459
4.6 Geometric flows 462
5 Construction of geometric structures on bundles and manifolds 463
5.1 Geometric structures with noncompact holonomy group 464
5.2 Uniformization for three manifolds 466
5.3 Four manifolds 469
5.4 Special connections on bundles 470
5.5 Symplectic structures 471
5.6 K？hler structure 474
5.7 Manifolds with special holonomy group 480
5.8 Geometric structures by reduction 480
5.9 Obstruction for existence of Einstein metrics on general manifolds 481
5.10 Metric Cobordism 481
References 482
A Survey of Calabi-Yau Manifolds&Shing-Tung Yau 521
1 Introduction 521
2 General Constructions of Complete Ricci-Flat Metrics in K？hler Geometry 521
2.1 The Ricci tensor of Calabi-Yau manifolds 521
2.2 The Calabi conjecture 522
2.3 Yau's theorem 522
2.4 Calabi-Yau manifolds and Calabi-Yau metrics 523
2.5 Examples of compact Calabi-Yau manifolds 524
2.6 Noncompact Calabi-Yau manifolds 525
2.7 Calabi-Yau cones:Sasaki-Einstein manifolds 526
2.8 The balanced condition on Calabi-Yau metrics 527
3 Moduli and Arithmetic of Calabi-Yau Manifolds 528
3.1 Moduli of K3 surfaces 528
3.2 Moduli of high dimensional Calabi-Yau manifolds 529
3.3 The modularity of Calabi-Yau threefolds over？ 530
4 Calabi-Yau Manifolds in Physics 531
4.1 Calabi-Yau manifolds in string theory 531
4.2 Calabi-Yau manifolds and mirror symmetry 532
4.3 Mathematics inspired by mirror symmetry 533
5 Invariants of Calabi-Yau Manifolds 534
5.1 Gromov-Witten Invariants 534
5.2 Counting formulas 534
5.3 Proofs of counting formulas for Calabi-Yau threefolds 535
5.4 Integrability of mirror map and arithmetic applications 536
5.5 Donaldson-Thomas invariants 537
5.6 Stable bundles and sheaves 538
5.7 Yau-Zaslow formula for K3 surfaces 539
5.8 Chern-Simons knot invariants,open strings and string dualities 540
6 Homological Mirror Symmetry 541
7 SYZ geometric interpretation of mirror symmetry 542
7.1 Special Lagrangian submanifolds in Calabi-Yau manifolds 542
7.2 The SYZ conjecture-SYZ transformation 543
7.3 Special Lagrangian geometry 543
7.4 Special Lagrangian fibrations 544
7.5 The SYZ transformation 545
7.6 The SYZ conjecture and tropical geometry 545
8 Geometries Related to Calabi-Yau Manifolds 546
8.1 Non-K？hler Calabi-Yau manifolds 546
8.2 Symplectic Calabi-Yau manifolds 547
References 548