Table of Contents

Preliminaries: Differential Forms, Exterior Derivative, Laplace Operator, Hodge Operator; Complex Manifolds: Complex Vector Fields, Differential Forms, Compatible Metrics, Blowing Up Points; Holomorphic Vector Bundles: Dolbeault Cohomology, Chern Connection, Some Formulas, Holomorphic Line Bundles; Kahler Manifolds: Kahler Form and Volume, Levi-Civita Connection, Curvature Tensor, Ricci Tensor, Holonomy, Killing Fields; Cohomology of Kahler Manifolds: Lefschetz Map and Differentials, Lefschetz Map and Cohomology, The $dd_c$-Lemma and Formality, Some Vanishing Theorems; Ricci Curvature and Global Structure: Ricci-Flat Kahler Manifolds, Nonnegative Ricci Curvature, Ricci Curvature and Laplace Operator; Calabi Conjecture: Uniqueness, Regularity, Existence, Obstructions; Kahler Hyperbolic Spaces: Kahler Hyperbolicity and Spectrum, Non-Vanishing of Cohomology; Kodaira Embedding Theorem: Proof of the Embedding Theorem, Two Applications; Appendix A: Chern-Weil Theory Appendix B: Symmetric Spaces Appendix C: Remarks on Differential Operators