Contents -- Preface -- 1 Background material -- 1.1 Exterior forms on manifolds -- 1.2 Introduction to analysis -- 1.3 Introduction to elliptic operators -- 1.4 Regularity of solutions of elliptic equations -- 1.5 Existence of solutions of linear elliptic equations -- 2 Introduction to connections, curvature and holonomy groups -- 2.1 Bundles, connections and curvature -- 2.2 Vector bundles, connections and holonomy groups -- 2.3 Holonomy groups and principal bundles -- 2.4 Holonomy groups and curvature -- 2.5 Connections on the tangent bundle, and torsion -- 2.6 G-structures and intrinsic torsion -- 3 Riemannian holonomy groups -- 3.1 Introduction to Riemannian holonomy groups -- 3.2 Reducible Riemannian manifolds -- 3.3 Riemannian symmetric spaces -- 3.4 The classification of Riemannian holonomy groups -- 3.5 Holonomy groups, exterior forms and cohomology -- 3.6 Spinors and holonomy groups -- 4 Calibrated geometry -- 4.1 Minimal submanifolds and calibrated submanifolds -- 4.2 Calibrated geometry and Riemannian holonomy groups -- 4.3 Classification of calibrations on R[sup(n)] -- 4.4 Geometric measure theory and tangent cones -- 5 Kähler manifolds -- 5.1 Introduction to complex manifolds -- 5.2 Tensors on complex manifolds -- 5.3 Holomorphic vector bundles -- 5.4 Introduction to Kähler manifolds -- 5.5 Kähler potentials -- 5.6 Curvature of Kähler manifolds -- 5.7 Exterior forms on Kähler manifolds -- 5.8 Complex algebraic varieties -- 5.9 Singular varieties, resolutions, and deformations -- 5.10 Line bundles and divisors -- 6 The Calabi Conjecture -- 6.1 Reformulating the Calabi Conjecture -- 6.2 Overview of the proof of the Calabi Conjecture -- 6.3 Calculations at a point -- 6.4 The proof of Theorem C1 -- 6.5 The proof of Theorem C2 -- 6.6 The proof of Theorem C3 -- 6.7 The proof of Theorem C4 -- 6.8 A discussion of the proof.

7 Calabi-Yau manifolds -- 7.1 Ricci-flat Kähler manifolds and Calabi-Yau manifolds -- 7.2 Crepant resolutions, small resolutions, and flops -- 7.3 Crepant resolutions of quotient singularities -- 7.4 Complex orbifolds -- 7.5 Crepant resolutions of orbifolds -- 7.6 Complete intersections -- 7.7 Deformations of Calabi-Yau manifolds -- 8 Special Lagrangian geometry -- 8.1 Special Lagrangian submanifolds in C[sup(m)] -- 8.2 Constructing examples of SL m-folds in C[sup(m)] -- 8.3 SL cones and Asymptotically Conical SL m-folds -- 8.4 SL m-folds in (almost) Calabi-Yau m-folds -- 8.5 SL m-folds with isolated conical singularities -- 9 Mirror symmetry and the SYZ Conjecture -- 9.1 String theory and mirror symmetry for dummies -- 9.2 Early mathematical formulations of mirror symmetry -- 9.3 Kontsevich's homological mirror symmetry proposal -- 9.4 The SYZ Conjecture -- 10 Hyperkähler and quaternionic Kähler manifolds -- 10.1 An introduction to hyperkähler geometry -- 10.2 Hyperkähler ALE spaces -- 10.3 K3 surfaces -- 10.4 Higher-dimensional compact hyperkähler manifolds -- 10.5 Quaternionic Kähler manifolds -- 10.6 Other topics in quaternionic geometry -- 11 The exceptional holonomy groups -- 11.1 The holonomy group G[sub(2)] -- 11.2 Topological properties of compact G[sub(2)]-manifolds -- 11.3 Constructing compact G[sub(2)]-manifolds -- 11.4 The holonomy group Spin(7) -- 11.5 Topological properties of compact Spin(7)-manifolds -- 11.6 Constructing compact Spin(7)-manifolds -- 11.7 Further reading on the exceptional holonomy groups -- 12 Associative, coassociative and Cayley submanifolds -- 12.1 Associative 3-folds and coassociative 4-folds in R[sup(7)] -- 12.2 Constructing associative and coassociative k-folds in R[sup(7)] -- 12.3 Associative 3- and coassociative 4-folds in G[sub(2)]-manifolds -- 12.4 Cayley 4-folds in R[sup(8)].

12.5 Cayley 4-folds in Spin(7)-manifolds -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W -- Y -- Z.