Compact Kähler Manifolds

Abstract

We summarize classical Hodge theory for compact Kähler manifolds and derive some important consequences. More precisely, in §1.1.1 we recall Hodge's Isomorphism Theorem for compact oriented Riemannian manifolds, stating that in any De Rhamcohomology class one can and a unique representative which is a harmonic form. This powerful theorem makes it possible to check various identities among cohomology classes on the level of forms. By definition a Kähler manifold is a complex hermitian manifold such that the associated metric form is closed and hence defines a cohomology class. The existence of such metrics has deep consequences. In §1.1.2 and §1.2.2 we treat this in detail, the highlights being the Hodge Decomposition Theorem and the Hard Lefschetz theorem. Here some facts about representation theory of SL(2; ℝ) are needed which, together with basic results needed in Chapt. 10, are gathered in §1.2.1.