| 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 kiihler manifolds 3.3 lefschetz theorems appendix 3.a formality of compact kahler manifolds 3.b susy for kiihler 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 hirzebrueh-riemann-roch theorem 5.2 kodaira vanishing theorem and applications 5.3 kodaira embedding theorem 6 deformations of complex structures 6.1 the maurer-caftan equation 6.2 general results appendix 6.a dgbv-algebras a hodge theory on differentiable manifolds b sheaf cohomology references index |
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