1 Preliminaries ................................................. 1
1.1 Differential Forms ........................................ 2
1.2 Exterior Derivative ....................................... 4
1.3 Laplace Operator .......................................... 6
1.4 Hodge Operator ............................................ 8
2 Complex Manifolds ............................................ 11
2.1 Complex Vector Fields .................................... 18
2.2 Differential Forms ....................................... 20
2.3 Compatible Metrics ....................................... 23
2.4 Blowing Up Points ........................................ 24
3 Holomorphic Vector Bundles ................................... 29
3.1 Dolbeault Cohomology ..................................... 29
3.2 Chern Connection ......................................... 31
3.3 Some Formulas ............................................ 34
3.4 Holomorphic Line Bundles ................................. 36
4 Kahler Manifolds ............................................. 41
4.1 Kahler Form and Volume ................................... 48
4.2 Levi-Civita Connection ................................... 50
4.3 Curvature Tensor ......................................... 52
4.4 Ricci Tensor ............................................. 54
4.5 Holonomy ................................................. 55
4.6 Killing Fields ........................................... 57
5 Cohomology of Kahler Manifolds ............................... 60
5.1 Lefschetz Map and Differentials .......................... 62
5.2 Lefschetz Map and Cohomology ............................. 65
5.3 The ddc -Lemma and Formality ............................. 71
5.4 Some Vanishing Theorems .................................. 75
6 Ricci Curvature and Global Structure ......................... 80
6.1 Ricci-Flat Kahler Manifolds .............................. 81
6.2 Nonnegative Ricci Curvature .............................. 82
6.3 Ricci Curvature and Laplace Operator ..................... 83
7 Calabi Conjecture ............................................ 86
7.1 Uniqueness ............................................... 89
7.2 Regularity ............................................... 91
7.3 Existence ................................................ 92
7.4 Obstructions ............................................. 97
8 Kahler Hyperbolic Spaces .................................... 103
8.1 Kahler Hyperbolicity and Spectrum ....................... 106
8.2 Non-Vanishing of Cohomology ............................. 110
9 Kodaira Embedding Theorem ................................... 114
9.1 Proof of the Embedding Theorem .......................... 116
9.2 Two Applications ........................................ 120
Appendix A. Chern-Weil Theory ................................. 121
A.l Chern Classes and Character ............................. 127
A.2 Euler Class ............................................. 131
Appendix B Symmetric Spaces ................................... 134
B.l Symmetric Pairs ......................................... 138
B.2 Examples ................................................ 144
B.3 Hermitian Symmetric Spaces .............................. 152
Appendix C. Remarks on Differential Operators ................. 157
C.l Dirac Operators ......................................... 160
C.2 L2-de Rham Cohomology ................................... 162
C.3 L2-Dolbeault Cohomology ................................. 163
Literature .................................................... 165
Index ......................................................... 171
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