| |
| |
Preface | |
| |
| |
| |
Background material | |
| |
| |
| |
Exterior forms on manifolds | |
| |
| |
| |
Introduction to analysis | |
| |
| |
| |
Introduction to elliptic operators | |
| |
| |
| |
Regularity of solutions of elliptic equations | |
| |
| |
| |
Existence of solutions of linear elliptic equations | |
| |
| |
| |
Introduction to connections, curvature and holonomy groups | |
| |
| |
| |
Bundles, connections and curvature | |
| |
| |
| |
Vector bundles, connections and holonomy groups | |
| |
| |
| |
Holonomy groups and principal bundles | |
| |
| |
| |
Holonomy groups and curvature | |
| |
| |
| |
Connections on the tangent bundle, and torsion | |
| |
| |
| |
G-structures and intrinsic torsion | |
| |
| |
| |
Riemannian holonomy groups | |
| |
| |
| |
Introduction to Riemannian holonomy groups | |
| |
| |
| |
Reducible Riemannian manifolds | |
| |
| |
| |
Riemannian symmetric spaces | |
| |
| |
| |
The classification of Riemannian holonomy groups | |
| |
| |
| |
Holonomy groups, exterior forms and cohomology | |
| |
| |
| |
Spinors and holonomy groups | |
| |
| |
| |
Calibrated geometry | |
| |
| |
| |
Minimal submanifolds and calibrated submanifolds | |
| |
| |
| |
Calibrated geometry and Riemannian holonomy groups | |
| |
| |
| |
Classification of calibrations on R[superscript n] | |
| |
| |
| |
Geometric measure theory and tangent cones | |
| |
| |
| |
Kahler manifolds | |
| |
| |
| |
Introduction to complex manifolds | |
| |
| |
| |
Tensors on complex manifolds | |
| |
| |
| |
Holomorphic vector bundles | |
| |
| |
| |
Introduction to Kahler manifolds | |
| |
| |
| |
Kahler potentials | |
| |
| |
| |
Curvature of Kahler manifolds | |
| |
| |
| |
Exterior forms on Kahler manifolds | |
| |
| |
| |
Complex algebraic varieties | |
| |
| |
| |
Singular varieties, resolutions, and deformations | |
| |
| |
| |
Line bundles and divisors | |
| |
| |
| |
The Calabi Conjecture | |
| |
| |
| |
Reformulating the Calabi Conjecture | |
| |
| |
| |
Overview of the proof of the Calabi Conjecture | |
| |
| |
| |
Calculations at a point | |
| |
| |
| |
The proof of Theorem C1 | |
| |
| |
| |
The proof of Theorem C2 | |
| |
| |
| |
The proof of Theorem C3 | |
| |
| |
| |
The proof of Theorem C4 | |
| |
| |
| |
A discussion of the proof | |
| |
| |
| |
Calabi-Yau manifolds | |
| |
| |
| |
Ricci-flat Kahler manifolds and Calabi-Yau manifolds | |
| |
| |
| |
Crepant resolutions, small resolutions, and flops | |
| |
| |
| |
Crepant resolutions of quotient singularities | |
| |
| |
| |
Complex orbifolds | |
| |
| |
| |
Crepant resolutions of orbifolds | |
| |
| |
| |
Complete intersections | |
| |
| |
| |
Deformations of Calabi-Yau manifolds | |
| |
| |
| |
Special Lagrangian geometry | |
| |
| |
| |
Special Lagrangian submanifolds in C[superscript m] | |
| |
| |
| |
Constructing examples of SL m-folds in C[superscript m] | |
| |
| |
| |
SL cones and Asymptotically Conical SL m-folds | |
| |
| |
| |
SL m-folds in (almost) Calabi-Yau m-folds | |
| |
| |
| |
SL m-folds with isolated conical singularities | |
| |
| |
| |
Mirror symmetry and the SYZ Conjecture | |
| |
| |
| |
String theory and mirror symmetry for dummies | |
| |
| |
| |
Early mathematical formulations of mirror symmetry | |
| |
| |
| |
Kontsevich's homological mirror symmetry proposal | |
| |
| |
| |
The SYZ Conjecture | |
| |
| |
| |
Hyperkahler and quaternionic Kahler manifolds | |
| |
| |
| |
An introduction to hyperkahler geometry | |
| |
| |
| |
Hyperkahler ALE spaces | |
| |
| |
| |
K3 surfaces | |
| |
| |
| |
Higher-dimensional compact hyperkahler manifolds | |
| |
| |
| |
Quaternionic Kahler manifolds | |
| |
| |
| |
Other topics in quaternionic geometry | |
| |
| |
| |
The exceptional holonomy groups | |
| |
| |
| |
The holonomy group G[subscript 2] | |
| |
| |
| |
Topological properties of compact G[subscript 2]-manifolds | |
| |
| |
| |
Constructing compact G[subscript 2]-manifolds | |
| |
| |
| |
The holonomy group Spin(7) | |
| |
| |
| |
Topological properties of compact Spin(7)-manifolds | |
| |
| |
| |
Constructing compact Spin(7)-manifolds | |
| |
| |
| |
Further reading on the exceptional holonomy groups | |
| |
| |
| |
Associative, coassociative and Cayley submanifolds | |
| |
| |
| |
Associative 3-folds and coassociative 4-folds in R[superscript 7] | |
| |
| |
| |
Constructing associative and coassociative k-folds in R[superscript 7] | |
| |
| |
| |
Associative 3- and coassociative 4-folds in G[subscript 2]-manifolds | |
| |
| |
| |
Cayley 4-folds in R[superscript 8] | |
| |
| |
| |
Cayley 4-folds in Spin(7)-manifolds | |
| |
| |
References | |
| |
| |
Index | |