Contents -- Introduction -- CHAPTER 1 Smooth manifolds -- 1.1. Introduction -- 1.2. The tangent space -- 1.3. Vector fields -- 1.4. Exercises -- CHAPTER 2 Tensor fields on smooth manifolds -- 2.1. Exterior and tensor algebras -- 2.2. Tensor fields -- 2.3. Lie derivative of tensors -- 2.4. Exercises -- CHAPTER 3 The exterior derivative -- 3.1. Exterior forms -- 3.2. The exterior derivative -- 3.3. The Cartan formula -- 3.4. Integration -- 3.5. Exercises -- CHAPTER 4 Principal and vector bundles -- 4.1. Lie groups -- 4.2. Principal bundles -- 4.3. Vector bundles -- 4.4. Correspondence between principal and vector bundles -- 4.5. Exercises -- CHAPTER 5 Connections -- 5.1. Covariant derivatives on vector bundles -- 5.2. Connections on principal bundles -- 5.3. Linear connections -- 5.4. Pull-back of bundles -- 5.5. Parallel transport -- 5.6. Holonomy -- 5.7. Reduction of connections -- 5.8. Exercises -- CHAPTER 6 Riemannian manifolds -- 6.1. Riemannian metrics -- 6.2. The Levi-Civita connection -- 6.3. The curvature tensor -- 6.4. Killing vector fields -- 6.5. Exercises -- CHAPTER 7 Complex structures and holomorphic maps -- 7.1. Preliminaries -- 7.2. Holomorphic functions -- 7.3. Complex manifolds -- 7.4. The complexified tangent bundle -- 7.5. Exercises -- CHAPTER 8 Holomorphic forms and vector fields -- 8.1. Decomposition of the (complexified) exterior bundle -- 8.2. Holomorphic objects on complex manifolds -- 8.3. Exercises -- CHAPTER 9 Complex and holomorphic vector bundles -- 9.1. Holomorphic vector bundles -- 9.2. Holomorphic structures -- 9.3. The canonical bundle of CPm -- 9.4. Exercises -- CHAPTER 10 Hermitian bundles -- 10.1. The curvature operator of a connection -- 10.2. Hermitian structures and connections -- 10.3. Exercises -- CHAPTER 11 Hermitian and Kähler metrics -- 11.1. Hermitian metrics -- 11.2. Kähler metrics.

11.3. Characterization of Kähler metrics -- 11.4. Comparison of the Levi-Civita and Chern connections -- 11.5. Exercises -- CHAPTER 12 The curvature tensor of Kähler manifolds -- 12.1. The Kählerian curvature tensor -- 12.2. The curvature tensor in local coordinates -- 12.3. Exercises -- CHAPTER 13 Examples of Kähler metrics -- 13.1. The flat metric on Cm -- 13.2. The Fubini-Study metric on the complex projective space -- 13.3. Geometrical properties of the Fubini-Study metric -- 13.4. Exercises -- CHAPTER 14 Natural operators on Riemannian and Kähler manifolds -- 14.1. The formal adjoint of a linear di erential operator -- 14.2. The Laplace operator on Riemannian manifolds -- 14.3. The Laplace operator on Kähler manifolds -- 14.4. Exercises -- CHAPTER 15 Hodge and Dolbeault theories -- 15.1. Hodge theory -- 15.2. Dolbeault theory -- 15.3. Exercises -- CHAPTER 16 Chern classes -- 16.1. Chern-Weil theory -- 16.2. Properties of the first Chern class -- 16.3. Exercises -- CHAPTER 17 The Ricci form of Kähler manifolds -- 17.1. Kähler metrics as geometric Um-structures -- 17.2. The Ricci form as curvature form on the canonical bundle -- 17.3. Ricci-flat Kähler manifolds -- 17.4. Exercises -- CHAPTER 18 The Calabi-Yau theorem -- 18.1. An overview -- 18.2. Exercises -- CHAPTER 19 Kähler-Einstein metrics -- 19.1. The Aubin-Yau theorem -- 19.2. Holomorphic vector fields on Kähler-Einstein manifolds -- 19.3. Exercises -- CHAPTER 20 Weitzenböck techniques -- 20.1. The Weitzenböck formula -- 20.2. Vanishing results on Kähler manifolds -- 20.3. Exercises -- CHAPTER 21 The Hirzebruch-Riemann-Roch formula -- 21.1. Positive line bundles -- 21.2. The Hirzebruch-Riemann-Roch formula -- 21.3. Exercises -- CHAPTER 22 Further vanishing results -- 22.1. The Lichnerowicz formula for Kähler manifolds -- 22.2. The Kodaira vanishing theorem -- 22.3. Exercises.

CHAPTER 23 Ricci-flat Kähler metrics -- 23.1. Hyperkähler manifolds -- 23.2. Projective manifolds -- 23.3. Exercises -- CHAPTER 24 Explicit examples of Calabi-Yau manifolds -- 24.1. Divisors -- 24.2. Line bundles and divisors -- 24.3. Adjunction formulas -- 24.4. Exercises -- Bibliography -- Index.