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

$C^{k, \alpha}$       see “Holder space”
$C^{k}$ space      5
holonomy      54—55 227—239
holonomy, associative 3-fold      68 71 72 254—272
holonomy, coassociative 4-fold      68 71 72 254—272
holonomy, compact manifolds, Betti numbers of      238
holonomy, compact manifolds, constructing      233—239
holonomy, compact manifolds, moduli space of      232 253
holonomy, compact manifolds, topology of      230—232
holonomy, explicit metrics      253
holonomy, holonomy subgroups      230
holonomy, metrics Ricci-flat      229
holonomy, parallel spinors      230
instanton      253
, definition      227
-manifold      228
-manifold, with holonomy SU(2) or SU(3)      230
-structure      228 252
-structure, function $\Theta$       228
-structure, nearly parallel      253
-structure, positive 3-form      228 253
-structure, positive 4-form      228
-structure, small torsion      233 235
-structure, splitting of forms      229 231
-structure, torsion      228
-structure, torsion-free      228 233 236
      see “Lebesgue space”
      see “Sobolev space”
$\mathbb{H}$       see “Quaternions”
$\mathbb{O}$       see “Octonions”
*      see “Hodge star”
3-Sasakian manifold      222—223
3-Sasakian quotient construction      225
A-genus      64 216 242
A-model      184
Affine structure      168 194
Affine structure, integral      195 197 198
Age grading      131—133
ALE manifold, Calabi — Yau      234 250—251
ALE manifold, Eguchi — Hanson space      205—206 212 234 246
ALE manifold, hyperkahler      201 205—207
ALE manifold, Spin(7)      250—251
Almost Calabi — Yau m-folds      165—166
Almost Calabi — Yau m-folds, generic      166 173 177 199—200
Ambrose — Singer Holonomy Theorem      31 36
Associative 3-folds      68 71 72 254—272
Associative 3-folds, deformations      256
Associative 3-folds, examples      259—264 268—270
Associative 3-folds, ruled      261—262
Associative cones      258 260 263
Associative cones, and pseudoholomorphic curves in       258 261 262
Associative cones, as an integrable system      261
Associative cones, two-sided      258
Associative fibrations      272
Asymptotic cone      152
Asymptotically conical SL m-folds      152 161—165
Asymptotically conical SL m-folds, deformations      162
Asymptotically conical SL m-folds, examples      155—157 163—165
Asymptotically cylindrical manifold      237
Asymptotically Locally Euclidean      see “ALE manifold”
Atiyah — Singer index theorem      18 64 242 275
B-model      184
Berger's theorem      52
Betti number      2 238 248 252
Betti number, refined      59 231 243
Bianchi identities      34 41 43 56
Blow-up      93—94
Bochner theorem      60 125
Bootstrap method      117
Brane      180 183 184
Calabi Conjecture      54 85 100—121 211 234 237 239
Calabi — Yau manifold      54 68 85 100 122—145 239
Calabi — Yau manifold, A-model      184
Calabi — Yau manifold, almost      165—166 173 177 199—200
Calabi — Yau manifold, B-model      184
Calabi — Yau manifold, constructions      139—144
Calabi — Yau manifold, definition      126
Calabi — Yau manifold, deformations      144—145
Calabi — Yau manifold, Hodge numbers      126
Calabi — Yau manifold, mirror pair      143 180 183
Calabi — Yau orbifold      136
Calibrated geometry      65—74 146—177 254—277
Calibrated submanifold      67
Calibration      67 126
Calibration, classification on $\mathbb{R}^n$       69—72
Canonical bundle      96
Cartan — Kahler theory      147 153
Category      184—185
Category, abelian      185—186
Category, additive      185
Category, Ax-      187
Category, Calabi — Yau $A_{\infty ^-}$       190
Category, definition      184
Category, derived      186—187
Category, derived Fukaya      184 189
Category, dg-      187
Category, enhanced triangulated      190
Category, equivalence      184 185
Category, exact      185
Category, Fukaya      184 187—189
Category, functor      184
Category, linear      185
Category, ofcoherent sheaves      184 185 187
Category, triangulated      186—187
Category, triangulated $A_{\infty ^-}$       187 189—190
Cayley 4-folds      68 71 272—277
Cayley 4-folds and associative 3-folds      274
Cayley 4-folds and coassociative 4-folds      274
Cayley 4-folds and holomorphic surfaces      273
Cayley 4-folds and SL 4-folds      273
Cayley 4-folds and Spin(7) instantons      277
Cayley 4-folds, 2-ruled      274
Cayley 4-folds, deformations      273
Cayley 4-folds, examples      273—276
Cayley 4-folds, with isolated conical singularities      276
Cayley fibrations      276—277
Cayley integral current      277
Cayley numbers      see “Octonions”
Characteristic class      96
Characteristic class, A-genus      64 216 242
Characteristic class, first Chern class      96 99 100 123
Characteristic class, first Pontryagin class      232 242 244
Cheeger — Gromoll Theorem      61
Chow's theorem      92 97 126
Closed string mirror symmetry      181—183 190—191
Closed strings      180
Coassociative 4-folds      68 71 72 254—272
Coassociative 4-folds, 2-ruled      263—264
Coassociative 4-folds, asymptotically conical      263
Coassociative 4-folds, deformations      257
Coassociative 4-folds, examples      259—264 268—270
Coassociative 4-folds, with isolated conical singularities      270—271
Coassociative cones      258 261
Coassociative cones, 2-ruled      263
Coassociative fibrations      271—272
Coherent sheaf      184 185 187 190
Cohomology, de Rham      2
Cohomology, Dolbeault      81
Cohomology, Hodge numbers      87
Cohomology, ofsheaves      91 95
Cohomology, other cohomology theories      2
Cohomology, Poincare duality      2 131 132 134 168 232
Compact linear map      6 104
Complete intersection      140
Complex manifold      76—79
Complex manifold, biholomorphism      78
Complex manifold, holomorphic function      76
Complex manifold, holomorphic map      78
Complex manifold, rigid      94 133 140
Complex projective space      77

Complex projective space, weighted      134 135 141—142 181 190 249
Complex structure      76—77
Complex structure, almost      76
Complex symplectic manifold      214—216
Complex symplectic manifold, irreducible      214
Complex symplectic manifold, marked      215
Complex symplectic manifold, moduli space      215
Connection      19—24
Connection, Levi-Civita      39—41
Connection, on principal bundle      22
Connection, on tangent bundle      32—36
Connection, on vector bundle      21
Connection, torsion-free      34—36
Continuity method      106
Crepant resolution      127—133
Current      72
Current, calibrated      73
Current, Cayley      277
Current, integral      72
Current, rectifiable      72
Current, special Lagrangian      158 176
Curvature, and holonomy groups      30—32
Curvature, in principal bundles      23
Curvature, in vector bundles      21—22
Curvature, of Kahler metrics      84—85
Curvature, Ricci      42
Curvature, Riemann      40—43 45 50 52 56 58 84 112 125 229 232 235 236 244 247
Curvature, scalar      42
de Rham theorem      2
Deformation      94—95
Deformation, of $\mathbb{C}^{m}/G$       133
Deformation, of Calabi — Yau manifold      144—145
Deformation, smoothing      94
Deformation, universal      95
Deformation, versal      95
Derived category      186—187
Derived category, of coherent sheaves      184 187
Derived Fukaya category      169 184
Dg-category      187
Dirac operator      62—64 242 243
Discriminant      191
Discriminant, trivalent graph      196
Divisor      98—99
Divisor, exceptional      94 128
Divisor, prime      98
Dolbeault cohomology      81
Double point      128
Double point, ordinary      129
Double point, rational      130
Eguchi — Hanson space      205—206 212 234 246
Elliptic operator      7—18
Elliptic operator, estimate      13
Elliptic operator, definition      9 11
Elliptic operator, existence of solutions      16—18
Elliptic operator, kernel finite-dimensional      16
Elliptic operator, nonlinear      9 102
Elliptic operator, regularity      12—15 66 74 116 121 162 167 171 237 258 266
Elliptic operator, Schauder estimate      13—14
Elliptic operator, symbol      9 11
Embedding      65
Enhanced triangulated category      190
Equivalence of categories      185
Exterior differential systems      54 147 153 252 256 264 270 273
Exterior form      see “Form”
Fano variety      180 190 221
Floer homology, Lagrangian      189
Floer homology, obstructions      169 188—189
FLOP      129
FORM      1—4
Form, complex symplectic      203
Form, G-structure splitting      56—58
Form, Hermitian      82
Form, holomorphic volume      123 126
Form, hyperkahler 2-form      203
Form, Kahler      82 123
Form, of type (p, q)      80
Form, on Kahler manifolds      85—86
Frobenius theorem      46
Fukaya category      169 184 187—189
Fukaya category, derived      189
functor      184 190
Functor, (left or right) derived      186
Functor, (left or right) exact      186
G-structure      36
Geometric measure theory      72—74 158 170 171 176 277
Green's representation      14
Gromov — Hausdorff limit      197
Gromov — Witten invariants      182—183 191
Hilbert scheme      218
HMS Conjecture      184 187—190
Hodge numbers      87
Hodge numbers, of hyperkahler manifolds      216
Hodge star      3 57
Hodge star, on Kahler manifolds      86
Hodge theory      3—4 58—60
Hodge theory, on Kahler manifolds      86—88
Holder space      5—6
Holder's inequality      4
Holomorphic vector bundle      81 185
Holomorphic volume form      126
Holonomy algebra      27 29 43
Holonomy group and cohomology      56—61
Holonomy group and curvature      30—32
Holonomy group, classification      52—56
Holonomy group, constant tensors      32—34
Holonomy group, definition      25 28 42
Holonomy group, exceptional      see “ $G_2$

holonomy” Holonomy “Spin(7) etc.
Holonomy group, for principal bundles      28—30
Holonomy group, for vector bundles      24—28
Holonomy group, restricted      26 28 42
Holonomy group, Ricci-flat      55
Holonomy group, Riemannian      42—43
Homological mirror symmetry      183—191
Homological mirror symmetry conjecture      see “HMS Conjecture”
Hypercomplex algebraic geometry      224
Hypercomplex manifold      223—224
Hypercomplex quotient construction      225
Hyperkahler manifold      54 100 201—219
Hyperkahler manifold, ALE      201 205—207
Hyperkahler manifold, cohomology      217
Hyperkahler manifold, examples      218—219
Hyperkahler manifold, Hodge numbers      216
Hyperkahler manifold, moduli space      217
Hyperkahler manifold, twistor space      204—205
Hyperkahler quotient construction      207 224
Hyperkahler structure      201 203
Hypersurface      98
Hypersurface, Calabi — Yau      140
Hypersurface, degree d      140
Hypersurface, in toric variety      143
Hypersurface, in weighted projective space      141—142
Immersion      65
Implicit Mapping Theorem      7
Injectivity radius      5 173 235 236 247
Instanton      225 253
Integrable systems      152—153 155 159 177 260—261
Integral affine structure      195 197 198
Integral current      72
Integral current, Cayley      277
Integral current, special Lagrangian      158 171 176
Interior estimate      15
Intrinsic torsion      38
Inverse mapping theorem      7 119
K3 surface      208—213
K3 surface, examples      208
K3 surface, marked      209
K3 surface, moduli space      209 213
Kahler chamber      210
Kahler class      84 88
Kahler cone      88 210 216

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