Introduction to Complex Manifolds

Cover Contents Preface Chapter 1. The Basics Definitions Holomorphic Maps Covering Manifolds and Quotient Manifolds Some Complex Analysis The Complexified Tangent and Cotangent Bundles Almost Complex Structures Problems Chapter 2. Complex Submanifolds Variations on the Inverse Function Theorem Complex Submanifolds Complex Submanifolds of Projective Spaces The Holomorphic Embedding Problem Problems Chapter 3. Holomorphic Vector Bundles Holomorphic Bundle Tools Holomorphic Line Bundles Line Bundles over Projective Space Applications of Holomorphic Line Bundles Problems Chapter 4. The Dolbeault Complex Decomposing Differential Forms by Type A Poincaré Lemma for the Dolbeault Operator Bundle-Valued Forms Problems Chapter 5. Sheaves Definitions The Étalé Space of a Presheaf Exact Sequences of Sheaves Problems Chapter 6. Sheaf Cohomology Definitions The Long Exact Cohomology Sequence Acyclic Resolutions Sheaf Cohomology and Singular Cohomology Applications of Sheaf Cohomology Other Sheaf Cohomology Theories Problems Chapter 7. Connections Connections on Complex Vector Bundles Curvature The First Real Chern Class The Chern Connection Problems Chapter 8. Hermitian and Kähler Manifolds Hermitian Metrics on the Tangent Bundle Kähler Metrics Examples of Kähler Metrics Curvature of Kähler Metrics Ricci and Scalar Curvatures Problems Chapter 9. Hodge Theory The Hodge Inner Product Elliptic Differential Operators Hodge Theory on Riemannian Manifolds Hodge Theory on Complex Manifolds Hodge Theory on Kähler Manifolds Applications of Hodge Theory Problems Chapter 10. The Kodaira Embedding Theorem Preliminaries The Kodaira Vanishing Theorem Proof of the Embedding Theorem Applications of the Embedding Theorem Problems Bibliography Notation Index Subject Index Back Cover