Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has re- cently obtained new impetus from physicists' interest in questions related to mirror symmetry. Due to its interactions with various other fields (differential, algebraic, and arithmetic geometry, but also string theory and conformal field theory), it has become an area with many facets. Also, there are a number of challenging open problems which contribute to the subject's attraction. The most famous among them is the Hodge conjecture, one of the seven one-million dollar millennium problems of the Clay Mathematics Institute. So, it seems likely t at this area will fascinate new generations for many years to come.

1 Local Theory

 1.1 Holomorphic Functions of Several Variables

 1.2 Complex and Hermitian Structures

 1.3 Differential Forms

2 Complex Manifolds

 2.1 Complex Manifolds: Definition and Examples

 2.2 Holomorphic Vector Bundles

 2.3 Divisors and Line Bundles

 2.4 The Projective Space

 2.5 Blow-ups

 2.6 Differential Calculus on Complex Manifolds

3 Kahler Manifolds

 3.1 Kahler Identities

 3.2 Hodge Theory on Kahler Manifolds

 3.3 Lefschetz Theorems

 Appendix

 3.A Formality of Compact Kahler Manifolds

 3.B SUSY for Kahler Manifolds

 3.C Hodge Structures

4 Vector Bundles

 4.1 Hermitian Vector Bundles and Serre Duality

 4.2 Connections

 4.3 Curvature

 4.4 Chern Classes

 Appendix

 4.A Levi-Civita Connection and Holonomy on Complex Manifolds

 4.B Hermite-Einstein and Kahler-Einstein Metrics

5 Applications of Cohomology

 5.1 Hirzebruch-Riemann-Roch Theorem

 5.2 Kodaira Vanishing Theorem and Applications

 5.3 Kodaira Embedding Theorem

 Deformations of Complex Structures

 6.1 The Maurer-Cartan Equation

 6.2 General Results

 Appendix

 6.A dGBV-Algebras

A Hodge Theory on Differentiable Manifolds

B Sheaf Cohomology

References

Index