Lectures on Kähler ManifoldsEuropean Mathematical Society, 2006 - 172 pagina's These 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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adjoint automorphic biholomorphic canonical Chern class Chern connection closed complex manifold closed Kähler manifold cohomology class compact complex manifold complex structure complex vector bundle Corollary corresponding curvature tensor decomposition defined denotes differential forms dmax equation example Exercise form of type forms with values function geodesic grad harmonic forms Hence Hermitian metric Hodge holomorphic coordinates holomorphic frame holomorphic line bundle holomorphic section holomorphic vector bundle holonomy induced invariant isometry isomorphism Kähler form Kähler manifold Kähler metric Kähler-Einstein metrics ker dmax Killing field Lefschetz Lemma Let G Lie algebra Lie group Math metric g non-compact type numbers open subset parallel positive Proof Proposition Remark respect Ricci curvature Riemannian manifold Riemannian metric Riemannian symmetric pair sectional curvature simply connected Sp(n submanifold symplectic tangent Theorem trivial vanishing vector fields vector space