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