The Hodge Decomposition Theorem on Compact Kähler Manifolds

Abstract

In this thesis we study the basics of differential analysis on complex manifolds. On Kähler manifolds we show that ∆ = 2 \square = 2 \bar{\square} and a few more useful relations between operators. Then we prove the Lefschetz decomposition theorem for harmonic forms on a Kähler manifold and we prove the Hodge decomposition theorem on a compact Kähler manifold X, which claims that the de Rham cohomology group H_r (X, C) can be decomposed as a direct sum of all Dolbeault cohomology groups H_{p,q} (X) with p + q = r. As a corollary to the last theorem we obtain rela- tions between the Betti numbers b_{r} (X) and the Hodge numbers h_{p,q} (X), which put topological restrictions on Kähler manifolds.