1 Local Theory 1
1.1 Holomorphic Functions of Several Variables 1
1.2 Complex and Hermitian Structures 25
1.3 Differential Forms 42
2 Complex Manifolds 51
2.1 Complex Manifolds: Definition and Examples 52
2.2 Holomorphic Vector Bundles 66
2.3 Divisors and Line Bundles 77
2.4 The Projective Space 91
2.5 Blow-ups 98
2.6 Differential Calculus on Complex Manifolds 104
3 Kahler Manifolds 113
3.1 Kahler Identities 114
3.2 Hodge Theory on Kahler Manifolds 125
3.3 Lefschetz Theorems 132
Appendix 145
3.A Formality of Compact Kahler Manifolds 145
3.B SUSY for Kahler Manifolds 155
3.C Hodge Structures 160
4 Vector Bundles 165
4.1 Hermitian Vector Bundles and Serre Duality 166
4.2 Connections 173
4.3 Curvature 182
4.4 Chern Classes 193
Appendix 206
4.A Levi-Civita Connection and Holonomy on Complex Manifolds . 206
4.B Hermite-Einstein and Kahler-Einstein Metrics 217
5 Applications of Cohomology 231
5.1 Hirzebruch-Riemann-Roch Theorem 231
5.2 Kodaira Vanishing Theorem and Applications 239
5.3 Kodaira Embedding Theorem 247
6 Deformations of Complex Structures 255
6.1 The Maurer-Cartan Equation 255
6.2 General Results 268
Appendix 275
6.A dGBV-Algebras 275
A Hodge Theory on Differentiate Manifolds 281
B Sheaf Cohomology 287
References 297
Index 303
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