Joyce D.D. — Compact manifolds with special holonomy :: Электронная библиотека попечительского совета мехмата МГУ

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Joyce D.D. — Compact manifolds with special holonomy

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Название: Compact manifolds with special holonomy

Автор: Joyce D.D.

Аннотация:

The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkahler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 447

Добавлена в каталог: 24.11.2013

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Предметный указатель

      see Hodge star
$C^{k,\alpha}(M)$       see H $\ddot{o}$ lder space
$C^{k}$ space      5
-structure      38
-manifold      243
-manifold, product      274
-manifold, QALE      272—278
-manifold, with holonomy SU(2)      271
-manifold, with holonomy SU(3)      272
-orbifolds $T^7 / \Gamma$       278—280
-orbifolds $T^7 / \Gamma$ , examples      312 315 316 329 334—337
-orbifolds $T^7 / \Gamma$ , R-data      280—281
-orbifolds $T^7 / \Gamma$ , resolutions of $T^7 / \Gamma$       282—284
-structure      243
-structure, function $\Theta$       243 249
-structure, nearly parallel      269
-structure, positive 3-form      243
-structure, positive 4-form      243
-structure, small torsion      284 286
-structure, splitting of forms      244 245
-structure, torsion      243
-structure, torsion-free      243
$G_{2}$ holonomy      57—58 242—254
$G_{2}$ holonomy, associative 3-fold      70 264—265 326—327 417
$G_{2}$ holonomy, coassociative 4-fold      70 265—267 327—328 417
$G_{2}$ holonomy, compact manifolds, Betti numbers of      314 315 323 325 326 332—334 339
$G_{2}$ holonomy, compact manifolds, constructions      294—295 303—305
$G_{2}$ holonomy, compact manifolds, examples      309—326 329—338
$G_{2}$ holonomy, compact manifolds, finding $\pi_{1} (M)$       307—308
$G_{2}$ holonomy, compact manifolds, finding Betti numbers      308—309
$G_{2}$ holonomy, compact manifolds, moduli space of      251—254
$G_{2}$ holonomy, compact manifolds, topology of      245—247 306—309
$G_{2}$ holonomy, compact manifolds, with $\pi_{1}(M)\neq (1)$       314—315
$G_{2}$ holonomy, holonomy subgroups      245
$G_{2}$ holonomy, metrics Ricci-flat      244
$G_{2}$ holonomy, parallel spinors      245
$G_{2}$ holonomy, research problems      303—305 416—418
$G_{2}$ , definition      242
surface      155—162
surface, examples      156
surface, marked      157
surface, moduli space      157 161
      see Lebesgue space
$L^{P}_{k}(M)$       see Sobolev space
$\hat{A}$ -genus      67 164 259
$\mathbb{H}$       see Quaternions
$\mathbb{O}$       see Octonions
Age grading      130—132 200
ALE manifold      173—174
ALE manifold, analysis on      178—182
ALE manifold, asymptotic coordinates      174
ALE manifold, Calabi conjecture for      186—188
ALE manifold, Eguchi — Hanson space      153—154 160
ALE manifold, examples      177—178 393—394
ALE manifold, Hodge theory      183—184
ALE manifold, holonomy Spin(7)      345—346
ALE manifold, hyperk $\ddot{a}$ hler      148 153—155
ALE manifold, K $\ddot{a}$ hler      174 201
ALE manifold, radius function      174 175
ALE manifold, Ricci-flat K $\ddot{a}$ hler      175—178
ALE manifold, structure group Spin(7)      344 393—394
ALE manifold, weighted H $\ddot{o}$ lder space      176 179
ALE manifold, weighted Sobolev space      178
Ambrose — Singer Holonomy Theorem      32 37
Associative 3-fold      70 264—265 417
Associative 3-fold, examples      326—327
Asymptotically Locally Euclidean      see ALE manifold
Atiyah — Singer index theorem      19 67 259
Berger’s Theorem      55
Betti number      2
Betti number, refined      62 245 260
Bianchi identities      36 43 45 59
Blow-up      91—92 293
Bochner theorem      64 124
Bootstrap method      116 303 364
Calabi Conjecture      98—100
Calabi conjecture, for ALE manifolds      186—188
Calabi conjecture, for QALE manifolds      225 229
Calabi — Yau manifold      56 70 82 98 121—147 304
Calabi — Yau manifold, constructions      138—142 146
Calabi — Yau manifold, definition      123
Calabi — Yau manifold, deformations      142—144
Calabi — Yau manifold, Hodge numbers      125
Calabi — Yau orbifold      135 146 395—396 401-
Calibrated geometry      68—70 264—269
Calibrated submanifold      68
Calibration      68
Canonical bundle      94
Cayley 4-fold      70 267—268 417
Cayley 4-fold, examples      371—373
Cayley numbers      see Octonions
Characteristic class      94
Characteristic class, $\hat{A}$ -genus      67 164 259
Characteristic class, first Chern class      94 97 98 122
Characteristic class, first Pontryagin class      246—247 259 261
Cheeger — Gromoll Theorem      64
Chow’s theorem      90 95 126
Coassociative 4-fold      70 265—267 417
Coassociative 4-fold, examples      327—328
Cohomology, de Rham      2 182
Cohomology, Dolbeault      77
Cohomology, Hodge numbers      85
Cohomology, of sheaves      89 93
Cohomology, other cohomology theories      2
Cohomology, Poincar $\acute{e}$ duality      3
Cohomology, with compact support      182
Compact linear map      6 103 179 187 226
Complete intersection      139 402 413
Complex manifold      72—75
Complex manifold, biholomorphism      74
Complex manifold, holomorphic function      72
Complex manifold, holomorphic map      74
Complex manifold, rigid      92 132 139
Complex projective space      73
Complex structure      72—73
Complex structure, almost      72
Complex symplectic manifold      162—164
Complex symplectic manifold, irreducible      162
Complex symplectic manifold, marked      163
Complex symplectic manifold, moduli space      163
Connection      20—25
Connection, Levi — Civita      40 42—43
Connection, on principal bundle      23
Connection, on tangent bundle      33—38
Connection, on vector bundle      22
Connection, torsion-free      35—38
Continuity method      104
Crepant resolution      126—132
Crepant resolution, of $\mathbb{C}^{m}/G$       175 210
Curvature and holonomy groups      32—33
Curvature in principal bundles      24
Curvature in vector bundles      22—23
Curvature of K $\ddot{a}$ hler metrics      81—82
Curvature, Ricci      44
Curvature, Riemann      43—44
Curvature, scalar      44
de Rham theorem      2
Deformation      92—93
Deformation, of $\mathbb{C}^{m}/G$       132 201 238
Deformation, of Calabi — Yau manifold      142—144
Deformation, smoothing      92 313 318 324
Deformation, universal      93
Deformation, versal      93
Dirac operator      65—68 259 260
Divisor      96—97
Divisor, exceptional      91 127
Divisor, prime      96
Dolbeault cohomology      77
Douady space      166 215
Double point      127

Double point, ordinary      127
Double point, rational      129
Ebin’s Slice Theorem      252
Eguchi — Hanson space      153—154 160 177 212 311 313 370 374
Elliptic equation nonlinear      250
Elliptic operator      7—19
Elliptic operator, $L^{p}$ estimate      14
Elliptic operator, definition      9 11
Elliptic operator, existence of solutions      16—19
Elliptic operator, kernel finite-dimensional      17
Elliptic operator, nonlinear      10 100
Elliptic operator, regularity      13—16 297 303 364
Elliptic operator, Schauder estimate      14—15 180
Elliptic operator, symbol      9 11
Exterior form      see Form
Fano manifold      168
FLOP      128
FORM      1—4
Form, -structure splitting      59—60
Form, Hermitian      78
Form, holomorphic volume      122
Form, hyperk $\ddot{a}$ hler 2-form      151
Form, K $\ddot{a}$ hler      79 122
Form, of type       77
Form, on K $\ddot{a}$ hler manifolds      82—84
Form,complex symplectic      150
Frobenius theorem      48
Green’s representation      14 181
H $\ddot{o}$ lder space      5—6
H $\ddot{o}$ lder space, weighted      176 179 221
H $\ddot{o}$ lder’s inequality      4
Harmonic coordinates      298
Hilbert scheme      166 215
Hodge numbers      85
Hodge numbers, of hyperk $\ddot{a}$ hler manifolds      164
Hodge star      3 60
Hodge star, on K $\ddot{a}$ hler manifolds      83
Hodge theory      4 61—63
Hodge theory, on ALE manifolds      183—184
Hodge theory, on K $\ddot{a}$ hler manifolds      84—85
Holomorphic vector bundle      77
Holonomy algebra      28 30 45
Holonomy group, and cohomology      59—64
Holonomy group, and curvature      32—33
Holonomy group, classification      55—59
Holonomy group, constant tensors      34—35
Holonomy group, definition      26 29 45
Holonomy group, exceptional      see $G_2$

holonomy Spin(7) etc.
Holonomy group, for principal bundles      29—32
Holonomy group, for vector bundles      25—29
Holonomy group, restricted      28 29 45
Holonomy group, Ricci-flat      58
Holonomy group, Riemannian      44—45
Hypercomplex algebraic geometry      171 215
Hypercomplex manifold      148 167 170
Hyperk $\ddot{a}$ hler manifold      57 98 148—167 169
Hyperk $\ddot{a}$ hler manifold, ALE      148 153—155
Hyperk $\ddot{a}$ hler manifold, examples      165—167
Hyperk $\ddot{a}$ hler manifold, Hodge numbers      164
Hyperk $\ddot{a}$ hler manifold, moduli space      165
Hyperk $\ddot{a}$ hler manifold, QALE      211 213—215 239
Hyperk $\ddot{a}$ hler manifold, twistor space      151—152
Hyperk $\ddot{a}$ hler quotient      154 169
Hyperk $\ddot{a}$ hler structure      148 151
Hypersurface      96
Hypersurface, Calabi — Yau      139
Hypersurface, degree       139
Hypersurface, in toric variety      142
Hypersurface, in weighted projective space      140 401
Implicit Mapping Theorem      7 253 263
Injectivity radius      5 221 292—295 298 359- 399
Instanton      169 269
Interior estimate      16
Intrinsic torsion      40
Inverse mapping theorem      7 118 228
K $\ddot{a}$ hler chamber      158
K $\ddot{a}$ hler class      80 85 175 210
K $\ddot{a}$ hler cone      85 158 175 210
K $\ddot{a}$ hler form      79 122
K $\ddot{a}$ hler manifold      56
K $\ddot{a}$ hler metric      79
K $\ddot{a}$ hler potential      80
K $\ddot{a}$ hler potential type      215 229
Kodaira embedding theorem      96 125
Kondrakov Theorem      7 16 103
Kummer construction      137 156—157 160—161
Kuranishi family      94 143
Laplacian      3 7 10 14 61
Laplacian, on ALE manifolds      180—182
Laplacian, on K $\ddot{a}$ hler manifolds      83 105 198
Laplacian, on QALE manifolds      220—225
Lebesgue space      4—5
Lefschetz hyperplane theorem      97 140 404
Levi — Civita connection      40 42—43
Line bundle      94—96
Line bundle, ample      95 125
Line bundle, canonical      94 122 126
Line bundle, first Chern class      94 97 98 122
Line bundle, over $\mathbb{CP}^m$       94
Line bundle, positive      96
Line bundle, very ample      95
Maximum principle      16 222
McKay correspondence      129 131 154 309
Metric, Fubini — Study      53 80
Metric, Hermitian      78
Metric, hyperk $\ddot{a}$ hler      151
Metric, K $\ddot{a}$ hler      79
Metric, reducible      46—50
Metric, symmetric      50—55
Mirror Symmetry      142 145—146
Moduli space      63 169
Moduli space, of -manifolds      251—254
Moduli space, of surfaces      157 161
Moduli space, of complex symplectic manifolds      163
Moduli space, of hyperk $\ddot{a}$ hler manifolds      165
Moduli space, of Spin(7)-manifolds      261—264
Monge — Amp $\grave{e}$ re equation      100
Newlander — Nirenberg theorem      72
Nijenhuis tensor      72
Node      127
Octonions      57
Orbifold      132—136
Orbifold, Calabi — Yau      135 146 395—396 401^02
Orbifold, complex      133 166
Orbifold, crepant resolntion      136—139
Orbifold, group      133
Orbifold, K $\ddot{a}$ hler      134
Orbifold, of torus      137 278—280 346—347
Orbifold, point      133
Orbifold, real      133
Parallel transport      26
Period domain      158 163
Period domain, hyperk $\ddot{a}$ hler      161 165
Period map      143 157 163
Period map, hyperk $\ddot{a}$ hler      161 165
Picard group      94
Poincar $\acute{e}$ duality      3
Principal bundle      20
Principal bundle, connection      23
QALE manifold, analysis on      219—225
QALE manifold, Calabi conjeclure for      225 229
QALE manifold, definition      208 236—238
QALE manifold, examples      212—215 239—241 276—278
QALE manifold, generalized      235—241
QALE manifold, holonomy       277—278
QALE manifold, holonomy Sp()      211 213—215 239
QALE manifold, holonomy Spin(7)      345—346
QALE manifold, holonomy SU()      211—213 239
QALE manifold, K $\ddot{a}$ hler      208—210 238
QALE manifold, K $\ddot{a}$ hler potentials      215

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