| Introduction | p. ix |
| Basics of differential geometry | p. 1 |
| Smooth manifolds | p. 3 |
| Introduction | p. 3 |
| The tangent space | p. 4 |
| Vector fields | p. 6 |
| Exercises | p. 9 |
| Tensor fields on smooth manifolds | p. 13 |
| Exterior and tensor algebras | p. 13 |
| Tensor fields | p. 15 |
| Lie derivative of tensors | p. 17 |
| Exercises | p. 19 |
| The exterior derivative | p. 21 |
| Exterior forms | p. 21 |
| The exterior derivative | p. 21 |
| The Cartan formula | p. 23 |
| Integration | p. 24 |
| Exercises | p. 26 |
| Principal and vector bundles | p. 29 |
| Lie groups | p. 29 |
| Principal bundles | p. 31 |
| Vector bundles | p. 33 |
| Correspondence between principal and vector bundles | p. 33 |
| Exercises | p. 35 |
| Connections | p. 37 |
| Covariant derivatives on vector bundles | p. 37 |
| Connections on principal bundles | p. 39 |
| Linear connections | p. 41 |
| Pull-back of bundles | p. 41 |
| Parallel transport | p. 42 |
| Holonomy | p. 43 |
| Reduction of connections | p. 44 |
| Exercises | p. 45 |
| Riemannian manifolds | p. 47 |
| Riemannian metrics | p. 47 |
| The Levi-Civita connection | p. 48 |
| The curvature tensor | p. 49 |
| Killing vector fields | p. 51 |
| Exercises | p. 52 |
| Complex and Hermitian geometry | p. 55 |
| Complex structures and holomorphic maps | p. 57 |
| Preliminaries | p. 57 |
| Holomorphic functions | p. 59 |
| Complex manifolds | p. 59 |
| The complexified tangent bundle | p. 61 |
| Exercises | p. 62 |
| Holomorphic forms and vector fields | p. 65 |
| Decomposition of the (complexified) exterior bundle | p. 65 |
| Holomorphic objects on complex manifolds | p. 67 |
| Exercises | p. 68 |
| Complex and holomorphic vector bundles | p. 71 |
| Holomorphic vector bundles | p. 71 |
| Holomorphic structures | p. 72 |
| The canonical bundle of CP [superscript m] | p. 74 |
| Exercises | p. 75 |
| Hermitian bundles | p. 77 |
| The curvature operator of a connection | p. 77 |
| Hermitian structures and connections | p. 78 |
| Exercises | p. 80 |
| Hermitian and Kahler metrics | p. 81 |
| Hermitian metrics | p. 81 |
| Kahler metrics | p. 82 |
| Characterization of Kahler metrics | p. 83 |
| Comparison of the Levi-Civita and Chern connections | p. 85 |
| Exercises | p. 86 |
| The curvature tensor of Kahler manifolds | p. 87 |
| The Kahlerian curvature tensor | p. 87 |
| The curvature tensor in local coordinates | p. 88 |
| Exercises | p. 91 |
| Examples of Kahler metrics | p. 93 |
| The flat metric on C[superscript m] | p. 93 |
| The Fubini-Study metric on the complex projective space | p. 93 |
| Geometrical properties of the Fubini-Study metric | p. 95 |
| Exercises | p. 97 |
| Natural operators on Riemannian and Kahler manifolds | p. 99 |
| The formal adjoint of a linear differential operator | p. 99 |
| The Laplace operator on Riemannian manifolds | p. 100 |
| The Laplace operator on Kahler manifolds | p. 101 |
| Exercises | p. 104 |
| Hodge and Dolbeault theories | p. 105 |
| Hodge theory | p. 105 |
| Dolbeault theory | p. 107 |
| Exercises | p. 109 |
| Topics on compact Kahler manifolds | p. 111 |
| Chern classes | p. 113 |
| Chern-Weil theory | p. 113 |
| Properties of the first Chern class | p. 116 |
| Exercises | p. 118 |
| The Ricci form of Kahler manifolds | p. 119 |
| Kahler metrics as geometric U[subscript m]-structures | p. 119 |
| The Ricci form as curvature form on the canonical bundle | p. 119 |
| Ricci-flat Kahler manifolds | p. 121 |
| Exercises | p. 122 |
| The Calabi-Yau theorem | p. 125 |
| An overview | p. 125 |
| Exercises | p. 127 |
| Kahler-Einstein metrics | p. 129 |
| The Aubin-Yau theorem | p. 129 |
| Holomorphic vector fields on Kahler-Einstein manifolds | p. 131 |
| Exercises | p. 133 |
| Weitzenbock techniques | p. 135 |
| The Weitzenbock formula | p. 135 |
| Vanishing results on Kahler manifolds | p. 137 |
| Exercises | p. 139 |
| The Hirzebruch-Riemann-Roch formula | p. 141 |
| Positive line bundles | p. 141 |
| The Hirzebruch-Riemann-Roch formula | p. 142 |
| Exercises | p. 145 |
| Further vanishing results | p. 147 |
| The Lichnerowicz formula for Kahler manifolds | p. 147 |
| The Kodaira vanishing theorem | p. 149 |
| Exercises | p. 151 |
| Ricci-flat Kahler metrics | p. 153 |
| Hyperkahler manifolds | p. 153 |
| Projective manifolds | p. 155 |
| Exercises | p. 156 |
| Explicit examples of Calabi-Yau manifolds | p. 159 |
| Divisors | p. 159 |
| Line bundles and divisors | p. 161 |
| Adjunction formulas | p. 162 |
| Exercises | p. 165 |
| Bibliography | p. 167 |
| Index | p. 169 |
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