this book is an outgrowth and a considerable expansion of lectures given at Brandeis University in 1967-1968 and at Rice University in 1968-1969. The first four chapters are an attempt to survey in detail some recent developments in four somewhat different areas, of mathematics: geometry (manifolds and vector bundles), algebraic topology, differential geometry, and partial differential equations. In these chapters, I have developed various tools that are useful in the study of compact complex ma

Chapter I Manifolds and Vector Bundles
1. Manifolds
2. Vector Bundles
3. Almost Complex Manifolds and thecS-Operator
Chapter II Sheaf Theory
1. Presheaves and Sheaves
2. Resolutions of Sheaves
3. Cohomology Theory
Appendix A. Cech Cohomology with Coefficients in a Sheaf
ChaptercHI Differential Geometry
1. Hermitian Differential Geometry
2. ThecCanonical Connectioncand Curvaturecofca Hermitian Holomorphic Vector Bundle
3. Chern Classescof Differentiable Vector Bundles
4. Complex Line Bundles
Chapter IV Elliptic Operator Theory
1. Sobolev Spaces
2. Differential Operators
3. Pseudodifferential Operators
4. A Parametrixcfor Elliptic Differential Operators
5. Elliptic Complexes
Chapter V Compact Complex Manifolds
1. Hermitian Exterior Algebraconca Hermitian Vector Space
2. Harmonic Theorycon Compact Manifolds
3. Representations of st(2,cC) on Hermitian Exterior Algebras
4. Differential Operatorsconca Kahier Manifold
5. The Hodge Decomposition Theoremcon Compact Kahler Manifolds
6. ThecHodge-Riemann Bilinear Relationsconca Kahler Manifoldc
Chapter VI Kodaira's Projective Embedding Theorem
1. HodgeManifolds
2. Kodaira's Vanishing Theorem
3. Quadratic Transformations
4. Kodaira's Embedding Theorem
References
Subject Index
Author Index