## Lectures on Kähler ManifoldsThese notes are based on lectures the author gave at the University of Bonn and the Erwin Schrodinger Institute in Vienna. The aim is to give a thorough introduction to the theory of Kahler manifolds with special emphasis on the differential geometric side of Kahler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kahler manifolds. The more advanced topics are the cohomology of Kahler manifolds, Calabi conjecture, Gromov's Kahler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern-Weil theory, symmetric spaces, and $L^2$-cohomology. |

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### Contents

Contents | 1 |

Complex Manifolds II | 11 |

JXJY XY | 17 |

Holomorphic Vector Bundles | 29 |

Kahler Manifolds | 41 |

Then the associated differential twoform co defined | 51 |

Cohomology of Kahler Manifolds | 60 |

Ricci Curvature and Global Structure | 80 |

Calabi Conjecture | 86 |

### Common terms and phrases

adjoint Ar(M biholomorphic bilinear canonical Chern class Chern connection closed complex manifold closed Kahler manifold cohomology class compact compatible Riemannian metric complex dimension complex Lie group complex manifold complex structure complex submanifold complex vector bundle Corollary corresponding curvature tensor decomposition defined denotes differential forms dual endomorphisms equation example Exercise follows form of type forms with values formula geodesic harmonic forms Hence Hermitian connection Hermitian metric Hodge holomorphic frame holomorphic line bundle holomorphic map holomorphic section holomorphic vector bundle induced invariant isometry Kahler form Kahler manifold Kahler metric Killing field Lefschetz Lemma Let G Lie algebra Lie group metric g morphism numbers open subset parallel Proof Proposition pull back Remark respect Ricci curvature Riemannian manifold Riemannian metric Riemannian symmetric pair sectional curvature simply connected smooth section Sp(n subgroup subspace tangent Theorem trivial vanishing vector fields wedge product