Special Metrics and Group Actions in Geometry

Preface......Page 6
Contents......Page 8
About the Editors......Page 10
 1 Introduction......Page 12
  2.1 Construction of LVMB-Manifolds......Page 17
  2.2 Duality LVMB-Data  Fans and (Quasi)Lattices......Page 19
 3 Toric and LVMB Geometry......Page 21
 4 The Polytopal Case......Page 24
 5 A Variant of the Delzant Construction......Page 25
 References......Page 31
Homotopic Properties of Kähler Orbifolds......Page 33
 1 Introduction......Page 34
 2  Formality of Manifolds......Page 36
 3 Orbifolds......Page 40
 4 Elliptic Differential Operators on Orbifolds......Page 48
 5 Kähler Orbifolds......Page 52
  6.1 6-Dimensional Example......Page 55
  6.2 8-Dimensional Example......Page 58
 7 Simply Connected Sasakian Manifolds......Page 60
  7.1 A Simply Connected Non-formal Sasakian Manifold......Page 61
  7.2 Simply Connected Formal Sasakian Manifolds with b2=0......Page 62
  7.3 Non-formal Quasi-Regular Sasakian Manifolds with b1=0......Page 64
 References......Page 65
 1 Prospectus......Page 68
  2.1 Classical Surface Geometry......Page 69
  2.3 Geometry of K-Surfaces......Page 70
   2.3.1 Bäcklund Transformations......Page 71
   2.4.1 Flat Connections......Page 72
  2.6 Sym Formula......Page 74
  2.7 Parallel Sections and Bäcklund Transformations......Page 75
  2.8 Permutability......Page 78
  3.1 Classical Theory......Page 79
  3.2 Conformal Geometry (Rapid Introduction)......Page 81
  3.3 Isothermic Surfaces Reformulated......Page 82
  3.6 Parallel Sections and Darboux Transforms......Page 83
  3.7 Permutability......Page 84
  3.8 Discrete Isothermic Surfaces......Page 85
  3.9 Discrete Gauge Theory......Page 86
  3.10 Gauge Theory of Discrete Isothermic Surfaces......Page 87
 References......Page 88
 1 Introduction......Page 90
  2.1 Conical and Projective Special Kähler Manifolds......Page 92
  2.2 Conical and Projective Special Kähler Domains......Page 93
 3 Special Kähler Manifolds with Regular Boundary Behaviour......Page 95
  4.1 The Supergravity C-Map......Page 98
  4.2 The One-Loop Deformation......Page 99
  4.3 Globalization of the One-Loop Deformed Metric......Page 100
  5.1 Completeness of the One-Loop Deformation for Projective Special Kähler Manifolds with Regular Boundary Behaviour......Page 105
  5.2 Completeness of the One-Loop Deformation for Complete Projective Special Kähler Manifolds with Cubic Prepotential......Page 109
   5.2.1 Projective Special Real Geometry and the Supergravity R-Map......Page 110
   5.2.2 The Completeness Theorem......Page 112
 References......Page 114
 1 Introduction......Page 116
 2 Hypertoric Manifolds......Page 117
  2.1 Abelian Actions......Page 118
  2.2 Dimension Four......Page 119
  2.3 Construction of Hypertoric Manifolds......Page 120
  2.4 Classification of Complete Hypertoric Manifolds......Page 125
 3 Cuts and Modifications......Page 129
 4 Non-Abelian Moment Maps......Page 130
 5 Implosion......Page 134
 References......Page 135
 1 Introduction......Page 137
 2 The Twistor Space of an Even-Dimensional Riemannian Manifold......Page 139
  2.1 The Twistor Space of an Oriented Four-Dimensional Riemannian Manifold......Page 141
 4 The Energy Functional on Sections of the Twistor Space......Page 144
 5 The Atiyah-Hitchin-Singer and Eells-Salamon Almost Complex Structures as Harmonic Sections......Page 148
 6 The Second Fundamental Form of an Almost Hermitian Structure as a Map into the Twistor Space......Page 151
 7 The Atiyah-Hitchin-Singer and Eells-Salamon Almost Complex Structures as Harmonic Maps......Page 156
 8 Almost Hermitian Structures on 4-Manifolds That are Harmonic Maps......Page 160
  8.1 The Case of Integrable J......Page 161
  8.3 Examples [13]......Page 162
  8.5  Inoue Surfaces of Type S0......Page 165
 References......Page 166
 1 Introduction......Page 168
 2 Killing 2-Forms and Ambikähler Structures......Page 172
 3 Separation of Variables......Page 183
 4 The Ambitoric Ansatz......Page 189
 5 Ambikähler Structures of Calabi Type......Page 196
 6 The Decomposable Case......Page 204
 7 Example: The Sphere S 4 and Its Deformations......Page 205
 8 Example: Complex Ruled Surfaces......Page 210
 References......Page 212
Twistors, Hyper-Kähler Manifolds, and Complex Moduli......Page 213
 References......Page 220
 1 Introduction and Organization of the Paper......Page 221
 2 The Rotation Invariant Case......Page 223
  2.1 Complete Reinhardt Domains......Page 225
  2.2 The Taub-NUT Metric......Page 226
  2.3 Gradient Kähler–Ricci Solitons......Page 228
 3 Calabi\'s Inhomogeneous Kähler–Einstein Metric on Tubular Domains......Page 233
 4 The Symplectic Geometry of Hermitian Symmetric Spaces......Page 235
  4.1 The Basic Example: The First Cartan Domain......Page 236
  4.2 Symplectic Capacities and Gromov Width of Hermitian Symmetric Spaces......Page 238
 References......Page 244
 1 Introduction......Page 246
  1.2 Dimension 7: G=G2......Page 247
  1.3 Dimension 8: G=Spin(7)......Page 248
  1.4 Construction Via Evolution......Page 249
  2.2 SU(2)-Instantons......Page 251
  2.3 Flat R4......Page 252
  2.4 Eguchi-Hanson......Page 253
  2.5 Taub-NUT......Page 255
  3.2 G2-Instantons......Page 257
  3.3 Flat R7......Page 259
  3.4 Self-dual 2-Forms Over S4......Page 260
  3.5 The Spinor Bundle Over S3......Page 263
  4.2 Spin(7)-Instantons......Page 266
  4.3 Flat R8......Page 267
  4.4 The Spinor Bundle Over S4......Page 268
 References......Page 271
 1 Introduction......Page 273
 2 Special Classes of Strongly Gauduchon Metrics......Page 276
 3 Deformation Limits of Hermitian Manifolds......Page 284
 References......Page 292
 1 Introduction......Page 295
 2 Poincaré Polynomials......Page 297
 3 The Octonionic Kähler 8-Form......Page 301
 4 Cayley-Rosenfeld Planes......Page 303
 5 Essential Clifford Structures......Page 307
 References......Page 308
 Introduction and Acknowledgement......Page 310
 1 What Is Holonomy?......Page 311
  1.1 The 2-Sphere......Page 312
  1.2 Complex Projective Space......Page 313
  1.3 Hermitian Manifolds......Page 315
  2.1 First Act......Page 316
  2.2 Second Act......Page 318
  2.3 Third Act......Page 319
  3.1 Curvature Constraint......Page 320
  3.2 Three-forms and four-forms......Page 321
 4 Explicit Metrics in Different Dimensions......Page 322
  4.1 Resolving Conical Holonomy......Page 323
  4.2 Examples from the 4-Torus......Page 325
  4.3 Weak Holonomy......Page 327
  5.1 Kummer Surface......Page 329
  5.2 Joyce\'s Construction for G2......Page 330
  5.3 The Twenty-First Century Approach......Page 332
  5.4 Twisted Connect Sums......Page 334
  5.5 Concluding Remarks......Page 335
 References......Page 338