Joyce D.D. — Riemannian holonomy groups and calibrated Geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Joyce D.D. — Riemannian holonomy groups and calibrated Geometry

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Название: Riemannian holonomy groups and calibrated Geometry

Автор: Joyce D.D.

Аннотация:

This graduate level text covers an exciting and active area of research at the crossroads of several different fields in mathematics and physics. In mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in physics String Theory and Mirror Symmetry.
Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kahler structures suitable for beginning graduate students, the text covers seminal results such as Yau's proof of the Calabi Conjecture, and takes the reader all the way to the frontiers of current research in calibrated geometry, giving many open problems.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 320

Добавлена в каталог: 22.05.2008

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Предметный указатель

Kahler form      82 123
Kahler manifold      54 82
Kahler metric      82
Kahler potential      84
Kodaira embedding theorem      98 126
Kondrakov Theorem      6 16 104
Kummer construction      138 208—209 212—213
Kuranishi family      96 144
Lagrangian Floer homology      189
Lagrangian Floer homology, obstructions      169 188—189
Lagrangian submanifolds      147
Landau — Ginzburg model      180 190
Laplacian      3 7 9 13 58
Laplacian, on Kahler manifolds      86 107
Large complex structure limit      182 183 189 192—194 196
Large radius limit      182 183 189 192—194 196
Lebesgue space      4—5
Lefschetz hyperplane theorem      99 141
Levi-Civita connection      39—41
Line bundle      96—98
Line bundle, ample      97 126
Line bundle, canonical      96 123 127
Line bundle, first Chern class      96 99 100 123
Line bundle, over CFm      96
Line bundle, positive      98
Line bundle, very ample      97
Log scheme      198
M-theory      179 227 252 253
Maximum principle      15
McKay correspondence      130 132 206
Mean curvature vector      66
Metric, Fubini — Study      51 83
Metric, Hermitian      82
Metric, hyperkahler      203
Metric, Kahler      82
Metric, reducible      44—48
Metric, symmetric      48—52
Minimal submanifold      66
Mirror pair      143 183
Mirror pair of Calabi — Yau 3-folds      180 183
Mirror Symmetry      142 178—200
Mirror symmetry, closed string      181—183 190—191
Mirror symmetry, homological      183—191
Mirror symmetry, of Hodge numbers      181
Mirror symmetry, quantum      192
Moduli space      60 225
Moduli space, of -manifolds      232 253
Moduli space, of Calabi — Yau manifolds      144—145
Moduli space, of complex symplectic manifolds      215
Moduli space, of hyperkahler manifolds      217
Moduli space, of K3 surfaces      209 213
Moduli space, of Spin(7)-manifolds      245
Moment map      150 153
Monge — Ampere equation      102 197
monodromy      194
Newlander-Nirenberg Theorem      76
Nijenhuis tensor      76
Node      129
Novikov ring      189 190
Octonions      54
Open strings      180
Orbifold      133—137
Orbifold, Calabi — Yau      136 248—252
Orbifold, complex      134 218
Orbifold, crepant resolution      137—140
Orbifold, group      134
Orbifold, Kahler      135
Orbifold, of torus      138 233—237 245—247
Orbifold, point      134
Orbifold, quaternionic Kahler      222
Orbifold, real      134
Parallel transport      25
Period domain      210 215
Period domain, hyperkahler      213 218
Period map      145 209 215
Period map, hyperkahler      213 218
Picard group      96
Poincare duality      2 131 132 134 168 232
Principal bundle      19
Principal bundle, connection      22
Pseudoholomorphic curves in       258 261 262
Quantum mirror symmetry      192
Quasi-ALE manifold, Calabi-Yau      235 246
Quaternionic Kahler manifold      54 201 219—222
Quaternionic Kahler manifold, negative      220
Quaternionic Kahler manifold, positive      220
Quaternionic Kahler manifold, twistor space      220
Quaternionic Kahler quotient construction      225
Quaternionic manifold      224
Quaternionic quotient construction      225
Quaternions      54 148 202
Quotient constructions      224—225
Real Monge — Ampere equation      197
Rectifiable current      72
Reduction Theorem      29
Resolution      93
Resolution, crepant      127—133
Resolution, of $\mathbb{C}^m /G$       129—133
Resolution, small      128
Ricci curvature      42 85
Ricci curvature and 1-forms      60—61
Ricci form      85 100 124
Riemann curvature      40—43 45 50 52 56 58 84 112 125 229 232 235 236 244 247
Riemannian holonomy      see “Holonomy group”
Riemannian metric      see “Metric”
Scalar curvature      42
SCFT      180—183 187 192
SCFT, topological twisting      183
Schauder estimate      13—14
Scheme      92
Schlessinger Rigidity Theorem      133
Second fundamental form      66
Self-dual 4-manifold      224
Self-dual 4-manifold, Einstein      221
Sheaf      91
Sheaf, coherent      184 185 187 190
Sheaf, cohomology      91 95
Sheaf, invertible      127

Singularity, canonical      128
Singularity, crepant resolution      127—133
Singularity, deformation      94—95
Singularity, Kleinian      130
Singularity, of variety      92
Singularity, resolution      93
Singularity, terminal      128 131
SL m-fold      see “Special Lagrangian m-folds”
Smoothing      94
Sobolev embedding theorem      6 111 237
Sobolev space      4—5
Sp(m) holonomy      see “Hyperkahler manifold”
Sp(m) Sp(1) holonomy      see “Quaternionic Kahler manifold”
Sp(m) Sp(1) holonomy, affine structures on moduli space      168—169
Sp(m) Sp(1) holonomy, as (semi)stable objects      169
Sp(m) Sp(1) holonomy, asymptotically conical      152 155—157 161—165
Sp(m) Sp(1) holonomy, asymptotically conical, deformations      162
Sp(m) Sp(1) holonomy, boundary ofmoduli Sp(m) Sp(1) holonomy, space      176
Sp(m) Sp(1) holonomy, compact      166—169
Sp(m) Sp(1) holonomy, connected sums      175—176
Sp(m) Sp(1) holonomy, constructions      150—157
Sp(m) Sp(1) holonomy, deformations      148—149 166—167
Sp(m) Sp(1) holonomy, examples      155—157 159—161 163—165
Sp(m) Sp(1) holonomy, in (almost) Calabi-Yau m-folds      165—177
Sp(m) Sp(1) holonomy, in (almost) Calabi-Yau m-folds, examples      169—170
Sp(m) Sp(1) holonomy, index of singularities      176—177 199 200
Sp(m) Sp(1) holonomy, obstructions      167—168
Sp(m) Sp(1) holonomy, ruled      151—152
Sp(m) Sp(1) holonomy, with isolated conical singularities      74 170—177
Sp(m) Sp(1) holonomy, with isolated conical singularities, behaviour near singularities      171—172
Sp(m) Sp(1) holonomy, with isolated conical singularities, deformations      172—173
Sp(m) Sp(1) holonomy, with isolated conical singularities, desingularizing      173—175
Sp(m) Sp(1) holonomy, with phase $e^{i\psi}$       147
Special Lagrangian cones      150 158—161
Special Lagrangian cones, as an integrable system      152
Special Lagrangian cones, examples      155—157 159—161
Special Lagrangian fibrations      155 191—193 198—200
Special Lagrangian fibrations, discriminant      191
Special Lagrangian fibrations, singularities      198—200
Special Lagrangian fibrations, U(1)-invariant      198—199
Special Lagrangian integral current      158 176
Special Lagrangian m-folds      68 71 146—177 239
Spin geometry      61—64
Spin structure      62
Spin(7) holonomy      54—55 239—252
Spin(7) holonomy, Cayley 4-fold      68 272—277
Spin(7) holonomy, compact manifolds, Betti numbers of      248 252
Spin(7) holonomy, compact manifolds, constructing      245—252
Spin(7) holonomy, compact manifolds, moduli space of      245
Spin(7) holonomy, compact manifolds, topology of      242—245
Spin(7) holonomy, explicit metrics      253
Spin(7) holonomy, holonomy subgroups      241
Spin(7) holonomy, metrics Ricci-flat      241
Spin(7) holonomy, parallel spinors      241
Spin(7) instanton      253 277
Spin(7), definition      239
Spin(7)-manifold      240
Spin(7)-manifold, with holonomy SU(2), SU(3), etc.      241—242
Spin(7)-structure      240
Spin(7)-structure, admissible 4-form      240
Spin(7)-structure, harmonic      64
Spin(7)-structure, parallel      62 230 241
Spin(7)-structure, small torsion      246 249
Spin(7)-structure, spinor      62
Spin(7)-structure, splitting of forms      240 243
Spin(7)-structure, torsion      240
Spin(7)-structure, torsion-free      240 247 249
Stokes' theorem      2
String theory      178—180 227 253
Strings, closed      180
Strings, open      180
Strominger — Yau — Zaslow Conjecture      see “SYZ Conjecture”
SU(m) holonomy      see “Calabi — Yau manifold”
Submanifold, definition      65
Submanifold, embedded      65
Submanifold, immersed      65
Submanifold, minimal      66
Super-conformal field theory      see “SCFT”
Swann bundle      223
Symmetric space      48—52
Symmetric space, constant curvature      51
Symmetric space, definition      48
SYZ Conjecture      146 155 191—200 271 272
SYZ Conjecture, as a limiting statement      192—194 196—198
Tangent cone      73—74 158 171 177
Torelli Theorem, Global      210 216
Torelli Theorem, Local      210 215
Torelli Theorem, Weak      210
Toric geometry      130
Torsion      34
Torsion, of G-structure      38
Triangulated category      186—187
Triangulated category, equivalence      184
Triangulated category, nonfunctoriality of the cone      186
Ttriangulated $A_{\infty}$ -category      189—190
Twistor space      204—205 220 224
Variety      88—92
Variety, abstract      92
Variety, affine      89
Variety, algebraic      78—79 90
Variety, analytic      92
Variety, blowing up      93—94
Variety, deformation      94—95
Variety, morphism      90
Variety, projective      90
Variety, rational map      90
Variety, resolution      93
Variety, singular      92
Variety, toric      130
Vector bundle      19
Vector bundle, connection      21
Vector bundle, holomorphic      81
Vector bundle, line bundle      see “Line bundle”
Weighted projective space      134 135 141—142 181 190 249
Weitzenbock formula      58 125
Weyl group      206—207
Wolf space      221
Yukawa coupling      182—183
Zariski topology      89 90

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