几何与分析 第2卷 英文PDF|Epub|txt|kindle电子书版本网盘下载

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Part 3 Mathematical Physics,Algebraic Geometry and Other Topics3

The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties&Bohan Fang,Chiu-Chu Melissa Liu, David Treumann and Eric Zaslow3

1 Introduction3

1.1 Outline4

2 Mirror symmetry for toric manifolds4

2.1 Hori-Vafa mirror4

2.2 Categories in mirror symmetry5

2.3 Results to date8

3 T-duality9

3.1 Moment polytope9

3.2 Geometry of the open orbit10

3.3 Statement of symplectic results12

3.4 T-dual of an equivariant line bundle14

4 Microlocalization16

4.1 Algebraic preliminaries16

4.2 The cast of categories17

4.3 Fukaya-Oh theorem19

4.4 Building the equivalence20

4.5 Equivalence and the inverse functor21

4.6 Singular support and characteristic cycles22

4.7 Comments on technicalities23

4.8 Statement of results25

5 Coherent-constructible correspondence25

6 Examples28

6.1 Taking the mapping cone28

6.2 Toric Fano surfaces29

6.3 Hirzebruch surfaces29

References34

Superspace：a Comfortably Vast Algebraic Variety&T.Hübsch39

1 Introduction39

1.1 Basic ideas and definitions40

1.2 The traditional superspace42

2 Off-shell worldline supermultiplets44

2.1 Adinkraic supermultiplets45

2.2 Various hangings46

2.3 Projected supermultiplets48

2.4 Supermultiplets vs.superfields49

3 Superspace,by construction50

3.1 Superpartners of time50

3.2 A telescoping deformation structure55

3.3 Nontrivial superspace geometry59

3.4 Higher-dimensional spacetime62

4 The comfortably vast superspace63

References65

A Report on theYau-Zaslow Formula&Naichung Conan Leung69

1 Yau-Zaslow formula and its generalizations70

2 Yau-Zaslow approach72

3 Matching method72

4 Degeneration method74

5 Calabi-Yau threefold method77

6 Conclusions78

References79

Hermitian-Yang-Mills Connections on K？hler Manifolds&Jun Li81

1 Introduction81

1.1 Hermitian-Yang-Mills connections81

1.2 HYM connections lead to stable bundles83

1.3 Stable bundles and their moduli spaces85

1.4 Flat bundles and stable bundles on curves86

2 Donaldson-Uhlenbeck-Yau theorem86

2.1 Donaldson's proof for algebraic surfaces87

2.2 Uhlenbeck-Yau's proof for K？hler manifolds88

3 Hermitian-Yang-Mills connections on curves90

4 Hermitian-Yang-Mills connections on surfaces92

4.1 Extending DUY correspondence92

4.2 Stable topology of the moduli spaces94

4.3 Donaldson polynomial invariants95

5 HYM connections on high dimensional varieties97

5.1 Extending the DUY correspondence in high dimensions97

5.2 Donaldson-Thomas invariants98

6 Concluding remark99

References99

Additivity and Relative Kodaira Dimensions&Tian-Jun Li and Weiyi Zhang103

1 Introduction103

2 Kodaira Dimensions and fiber bundles104

2.1 kh for complex manifolds and Kt up to dimension 3105

2.2 Ks for symplectic 4-manifolds107

2.3 Additivity for a fiber bundle109

3 Embedded symplectic surfaces and relative Kod.dim.in dim.4112

3.1 Embedded symplectic surfaces and maximality112

3.2 The adjoint class Kω+[F]115

3.3 Existence and Uniqueness of relatively minimal model122

3.4 ks(M,ω,F)124

4 Relative Kod.dim.in dim.2 and fibrations over a surface127

4.1 kt(F,D),Riemann-Hurwitz formula and Seifert fibrations128

4.2 Lefschetz fibrations130

References133

Descendent Integrals and Tautological Rings of Moduli Spaces of Curves&Kefeng Liu and Hao Xu137

1 Introduction137

2 Intersection numbers and the Witten-Kontsevich theorem138

2.1 Witten-Kontsevich theorem139

2.2 Virasoro constraints141

3 The n-point function142

3.1 A recursive formula of n-point functions142

3.2 An effective recursion formulae of descendent integrals145

4 Hodge integrals146

4.1 Faber's algorithm146

4.2 Hodge integral formulae147

5 Higher Weil-Petersson volumes149

5.1 Generalization of Mirzakhani's recursion formula149

5.2 Recursion formulae of higher Weil-Petersson volumes151

6 Faber's conjecture on tautological rings152

6.1 The Faber intersection number conjecture153

6.2 Relations with n-point functions154

7 Dimension of tautological rings155

7.1 Ramanujan's mock theta functions156

7.2 Asymptotics of tautological dimensions158

8 Gromov-Witten invariants161

8.1 Universal equations of Gromov-Witten invariants162

8.2 Some vanishing identities163

9 Witten's r-spin numbers164

9.1 Generalized Witten's conjecture165

9.2 An algorithm for computing Witten's r-spin numbers166

Referenees168

A General Voronoi Summation Formula for GL(n,？)&Stephen D.Miller and Wilfried Schmid173

1 Introduction173

2 Automorphic Distributions178

3 Vanishing to infinite order189

4 Classical proof of the formula206

5 Adelic proof of the formula211

References223

Geometry of Holomorphic Isometries and Related Maps between Bounded Domains&Ngaiming Mok225

1 Examples of holomorphic isometries229

1.1 Examples of equivariant embeddings into the projective space229

1.2 Non-standard holomorphic isometries of the Poincaré disk into polydisks231

1.3 A non-standard holomorphic isometry of the Poincaré disk into a Siegel upper half-plane232

1.4 Examples of holomorphic isometries with arbitrary normalizing constants λ>1232

2 Analytic continuation of germs of holomorphic isometries234

2.1 Analytic continuation of holomorphic isometries into the projective space equipped with the Fubini-Study metric234

2.2 An extension and rigidity problem arising from commutators of modular correspondences236

2.3 Analytic continuation of holomorphic isometries up to normalizing constants with respect to the Bergman metric-extension beyond the boundary240

2.4 Canonically embeddable Bergman manifolds and Bergman meromorphic compactifications246

3 Holomorphic isometries of the Poincaré disk into bounded symmetric domains249

3.1 Structural equations on the norm of the second fundamental form and asymptotic vanishing order249

3.2 Holomorphic isometries of the Poincaré disk into polydisks:structural results250

3.3 Calculated examples on the norm of the second fundamental form251

3.4 Holomorphic isometries of the Poincaré disk into polydisks:uniqueness results253

3.5 Asymptotic total geodesy and applications254

4 Measure-preserving algebraic correspondences on irreducible bounded symmetric domains255

4.1 Statements of results255

4.2 Extension results on strictly pseudoconvex algebraic hypersurfaces256

4.3 Alexander-type extension results in the higher-rank case257

4.4 Total geodesy of germs of measure-preserving holomorphic map from an irreducible bounded symmetric domain of dimension≥2 into its Cartesian products259

5 Open problems261

5.1 On the structure of the space of holomorphic isometries of the Poincaré disk into polydisks261

5.2 On the second fundamental form and asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains264

5.3 On germs of holomorphic maps preserving invariant(p,p)-forms266

References267

Abundance Conjecture&Yum-Tong Siu271

0 Introduction271

1 Curvature current and dichotomy275

2 Gelfond-Schneider's technique of algebraic values of solutions of algebraically defined differential equations279

3 Final step of the case of zero numerical Kodaira dimension287

4 Numerically trivial foliations and fibrations for canonical line bundle289

5 Curvature of zeroth direct image of relative canonical and pluricanonical bundle291

6 Strict positivity of direct image of relative pluricanonical bundle along numerically trivial fibers in the base of numerically trivial fibration302

7 Technique of Nevanlinna's first main theorem for proof of compactness of leaves of foliation306

References315

Sasaki-Einstein Geometry&James Sparks319

1 Sasakian geometry319

2 Constructions of Sasaki-Einstein manifolds321

3 Obstructions324

4 Sasaki-Einstein manifolds in string theory326

References327

A Simple Proof of the Chiral Gravity Conjecture&Andrew Strominger329

Geometry of Grand Unification&Cumrun Vafa335

1 Introduction335

2 Standard model and gauge symmetry breaking336

3 Flavors and hierarchy337

4 Unification of gauge groups338

5 String theory,forces,matter,and interactions339

6 F-theory vacua340

6.1 Matter fields340

6.2 Yukawa couplings342

7 Applications to particle physics342

7.1 E-type singularity343

7.2 Flavor hierarchy343

7.3 Breaking to the standard model344

8 Further issues345

References345

Quantum Invariance Under Flop Transitions&Chin-Lung Wang347

1 Introduction347

2 Ordinary flops:Genus zero theory350

2.1 The canonical correspondence350

2.2 The case of simple flops [9]351

2.3 The topological defect352

2.4 The extremal functions353

2.5 Degeneration analysis355

2.6 The local models356

3 Calabi-Yau flops358

3.1 The basic setup358

3.2 I,P,J and their degrees358

3.3 The CY condition and the mirror map360

3.4 Example:Flops of type(P1,？(-7)),？(3)？(2))361

3.5 Proof of the main result in the example365

References369

The Problem Of Gauge Theory&Edward Witten371

1 Yang-Mills equations371

2 Classical phase space373

3 Quantization376

4 Nonperturbative approach379

5 Breaking of conformal invariance and the mass gap381

References382

Part 4 Appendices385

Shing-Tung Yau,a Manifold Man of Mathematics&Lizhen Ji and Kefeng Liu385

1 Childhood and early school education386

2 Middle school and college387

3 Graduate school390

4 Professional career392

5 Major contributions to mathematics395

6 Visits to China404

7 Research centers and mathematics institutes405

8 ICCM408

9 Conferences and popular mathematics programs409

10 Mathematics and Chinese literature410

11 Family,friends and students410

12 Summary415

References416

Perspectives on Geometric Analysis&Shing-Tung Yau417

1 History and contributors of the subject419

1.1 Founding fathers of the subject419

1.2 Modern Contributors421

2 Construction of functions in geometry422

2.1 Polynomials from ambient space423

2.2 Geometric construction of functions426

2.3 Functions and tensors defined by linear differential equations430

3 Mappings between manifolds and rigidity of geometric structures446

3.1 Embedding446

3.2 Rigidity of harmonic maps with negative curvature449

3.3 Holomorphic maps451

3.4 Harmonic maps from two dimensional surfaces and pseudoholomorphic curves452

3.5 Morse theory for maps and topological applications453

3.6 Wave maps454

3.7 Integrable system454

3.8 Regularity theory455

4 Submanifolds defined by variational principles455

4.1 Teichmüller space455

4.2 Classical minimal surfaces in Euclidean space456

4.3 Douglas-Morrey solution,embeddedness and application to topology of three manifolds457

4.4 Surfaces related to classical relativity458

4.5 Higher dimensional minimal subvarieties459

4.6 Geometric flows462

5 Construction of geometric structures on bundles and manifolds463

5.1 Geometric structures with noncompact holonomy group464

5.2 Uniformization for three manifolds466

5.3　 Four manifolds469

5.4　 Special connections on bundles470

5.5 　Symplectic structures471

5.6　 K？hler structure474

5.7 　Manifolds with special holonomy group480

5.8 Geometric structures by reduction480

5.9 Obstruction for existence of Einstein metrics on general manifolds481

5.10 Metric Cobordism481

References482

A Survey of Calabi-Yau Manifolds&Shing-Tung Yau521

1 Introduction521

2 General Constructions of Complete Ricci-Flat Metrics in K？hler Geometry521

2.1 The Ricci tensor of Calabi-Yau manifolds521

2.2 The Calabi conjecture522

2.3 Yau's theorem522

2.4 Calabi-Yau manifolds and Calabi-Yau metrics523

2.5 Examples of compact Calabi-Yau manifolds524

2.6 Noncompact Calabi-Yau manifolds525

2.7 Calabi-Yau cones:Sasaki-Einstein manifolds526

2.8 The balanced condition on Calabi-Yau metrics527

3 Moduli and Arithmetic of Calabi-Yau Manifolds528

3.1 Moduli of K3 surfaces528

3.2 Moduli of high dimensional Calabi-Yau manifolds529

3.3 The modularity of Calabi-Yau threefolds over？530

4 Calabi-Yau Manifolds in Physics531

4.1 Calabi-Yau manifolds in string theory531

4.2 Calabi-Yau manifolds and mirror symmetry532

4.3 Mathematics inspired by mirror symmetry533

5 Invariants of Calabi-Yau Manifolds534

5.1 Gromov-Witten Invariants534

5.2 Counting formulas534

5.3 Proofs of counting formulas for Calabi-Yau threefolds535

5.4 Integrability of mirror map and arithmetic applications536

5.5 Donaldson-Thomas invariants537

5.6 Stable bundles and sheaves538

5.7 Yau-Zaslow formula for K3 surfaces539

5.8 Chern-Simons knot invariants,open strings and string dualities540

6 Homological Mirror Symmetry541

7 SYZ geometric interpretation of mirror symmetry542

7.1 Special Lagrangian submanifolds in Calabi-Yau manifolds542

7.2 The SYZ conjecture-SYZ transformation543

7.3 Special Lagrangian geometry543

7.4 Special Lagrangian fibrations544

7.5 The SYZ transformation545

7.6 The SYZ conjecture and tropical geometry545

8 Geometries Related to Calabi-Yau Manifolds546

8.1 Non-K？hler Calabi-Yau manifolds546

8.2 Symplectic Calabi-Yau manifolds547

References548