Compact manifolds with special holonomy

Dominic D. Joyce0198506015, 9780198506010

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 This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.

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Table of contents :
Preface ……Page 6
Contents ……Page 8
1.1 Exterior forms on manifolds ……Page 12
1.2 Introduction to analysis ……Page 15
1.3 Introduction to elliptic operators ……Page 18
1.4 Regularity of solutions of elliptic equations ……Page 24
1.5 Existence of solutions of linear elliptic equations ……Page 27
2.1 Bundles, connections and curvature ……Page 31
2.2 Vector bundles, connections and holonomy groups ……Page 36
2.3 Holonomy groups and principal bundles ……Page 40
2.4 Holonomy groups and curvature ……Page 43
2.5 Connections on the tangent bundle, and torsion ……Page 44
2.6 $G$-structures and intrinsic torsion ……Page 49
3.1 Introduction to Riemannian holonomy groups ……Page 53
3.2 Reducible Riemannian manifolds ……Page 57
3.3 Riemannian symmetric spaces ……Page 61
3.4 The classification of Riemannian holonomy groups ……Page 66
3.5 Holonomy groups and de Rham cohomology ……Page 70
3.6 Spinors and holonomy groups ……Page 75
3.7 Calibrated geometry and holonomy groups ……Page 79
4 Kahler manifolds ……Page 82
4.1 Introduction to complex manifolds ……Page 83
4.2 Tensors on complex manifolds ……Page 86
4.3 Holomorphic vector bundles ……Page 88
4.4 Introduction to Kahler manifolds ……Page 89
4.5 Kahler potentials ……Page 91
4.6 Curvature of Kahler manifolds ……Page 92
4.7 Exterior forms on Kahler manifolds ……Page 93
4.8 Complex algebraic varieties ……Page 97
4.9 Singular varieties, resolutions, and deformations ……Page 101
4.10 Line bundles and divisors ……Page 105
5 The Calabi conjecture ……Page 109
5.1 Reformulating the Calabi conjecture ……Page 110
5.2 Overview of the proof of the Calabi conjecture ……Page 112
5.3 Calculations at a point ……Page 116
5.4 The proof ofTheorem C1 ……Page 119
5.5 The proof of Theorem C2 ……Page 126
5.6 The proof of Theorem C3 ……Page 128
5.8 A discussion of the proof ……Page 130
6 Calabi-Yau manifolds ……Page 132
6.1 The holonomy groups $SU(m)$ ……Page 133
6.2 Compact Ricci-flat Kahler manifolds ……Page 134
6.3 Crepant resolutions, small resolutions, and flops ……Page 137
6.4 Crepant resolutions of quotient singularities ……Page 139
6.5 Complex orbifolds ……Page 143
6.6 Crepant resolutions of orbifolds ……Page 147
6.7 Complete intersections ……Page 150
6.8 Deformations of Calabi-Yau manifolds ……Page 153
6.9 String Theory and Mirror Symmetry ……Page 155
6.10 Further reading on Calabi-Yau manifolds ……Page 157
7 Hyperkahler manifolds ……Page 159
7.1 An introduction to hyperkahler geometry ……Page 160
7.2 Hyperkahler ALE spaces ……Page 164
7.3 $K3$ surfaces ……Page 166
7.4 Higher-dimensional compact hyperkahler manifolds ……Page 173
7.5 The other quaternionic geometries ……Page 178
7.6 A reading list on quaternionic geometry ……Page 180
8 Asymptotically locally Euclidean metrics with holonomy $SU(m)$ ……Page 183
8.1 Introduction to ALE metrics ……Page 184
8.2 Ricci-flat ALE Kahler manifolds ……Page 186
8.3 Analysis on ALE manifolds ……Page 189
8.4 Exterior forms and de Rham cohomology ……Page 193
8.5 The Calabi conjecture for ALE manifolds ……Page 197
8.6 A priori estimates of $phi$ ……Page 199
8.7 The proofs of Theorems A1-A4 ……Page 207
8.8 The proofs of Theorems 8.2.3 and 8.2.4 ……Page 210
8.9 The case $m=2$, and deformations ……Page 212
9 QALE metrics with holonomy $SU(m)$ and $Sp(m)$ ……Page 214
9.1 Local product resolutions ……Page 215
9.2 Quasi-ALE Kahler metrics ……Page 219
9.3 Ricci-flat QALE Kahler manifolds ……Page 221
9.4 Kahler potentials on QALE Kahler manifolds ……Page 226
9.5 Analysis on QALE Kahler manifolds ……Page 230
9.6 The Calabi conjecture for QALE manifolds ……Page 236
9.7 A more complicated QALE Calabi conjecture ……Page 239
9.8 The proofs of Theorems 9.3.3 and 9.3.4 ……Page 243
9.9 Generalized QALE manifolds ……Page 246
10.1 The holonomy group $G_2$ ……Page 253
10.2 The topology of compact $G_2$-manifolds ……Page 256
10.3 Exterior forms on $G_2$-manifolds ……Page 258
10.4 The moduli space of holonomy Gi metrics ……Page 262
10.5 The holonomy group Spin(7) ……Page 265
10.6 The topology of compact Spin(7)-manifolds ……Page 270
10.7 The moduli space of holonomy Spin(7) metrics ……Page 272
10.8 Exceptional holonomy and calibrated geometry ……Page 275
10.9 A reading list on the exceptional holonomy groups ……Page 279
11 Construction of compact $G_2$-manifolds ……Page 281
11.1 Resolving $G_2$-singularities with holonomy SU(2), SU(3) ……Page 282
11.2 Quasi-ALE $G_2$-manifolds ……Page 285
11.3 Some notation to describe the orbifolds $T^7/Gamma$ ……Page 289
11.4 R-data and resolutions of $T^7/Gamma$ ……Page 291
11.5 $G_2$-structures on $M$ with small torsion ……Page 295
11.6 An existence result for compact $G_2$-manifolds ……Page 305
11.7 The proof of Theorem G1 ……Page 308
11.8 The proof of Theorem G2 ……Page 310
11.9 Other constructions of compact $G_2$-manifolds ……Page 314
12.1 Topological invariants of resolutions of $T^7/Gamma$ ……Page 317
12.2 A simple example ……Page 320
12.3 A modification of the example of §12.2 ……Page 323
12.4 Examples with nontrivial fundamental group ……Page 325
12.5 A more complicated example ……Page 326
12.6 Examples of associative and coassociative submanifolds ……Page 337
12.7 An example of a resolution of $T^7/Gamma$ with $Gamma$ nonabelian ……Page 340
12.8 More examples ……Page 345
12.9 Conclusions ……Page 349
13 Construction of compact Spin(7)-manifolds ……Page 352
13.1 Resolving Spin(7)-singularities and QALE Spin(7)-manifolds ……Page 353
13.2 Orbifolds $T^8/Gamma$ R-data, and resolutions of $T^8/Gamma$ ……Page 357
13.3 Construction of various exterior forms ……Page 360
13.4 Introducing a real parameter $tin (0,1]$ ……Page 363
13.5 Spin(7)-structures on $M$ with small torsion ……Page 366
13.6 An existence result for compact Spin(7)-manifolds ……Page 371
13.7 The proof of Theorem S2 ……Page 373
14.1 Topological invariants of resolutions of $T^8/Gamma$ ……Page 377
14.2 A simple example ……Page 379
14.3 Examples of Cayley submanifolds ……Page 382
14.4 A more complicated example ……Page 384
14.5 Another example with $Gamma=mathbb{Z}_2^4$ ……Page 391
14.6 An example with $Gamma$ nonabelian ……Page 397
15 A second construction of compact 8-manifolds with holonomy Spin(7) ……Page 403
15.1 ALE Spin(7)-manifolds ……Page 404
15.2 Proof of the construction ……Page 406
15.3 How to apply the construction ……Page 412
15.4 A simple example ……Page 417
15.5 Examples from hypersurfaces in $mathbb{CP}^5_{a_0,ldots,a_5}$ ……Page 419
15.6 A hypersurface of degree 8 in $mathbb{CP}^5_{1,1,1,1,2,2}$ over $mathbb{Z}_2$ ……Page 422
15.7 Examples from complete intersections in $mathbb{CP}^6_{a_0,ldots,a_6}$ ……Page 424
15.8 Discussion, and questions for future research ……Page 427
References ……Page 430
Index ……Page 443