Kähler Manifolds

Abstract

We come now to the heart of our story. We saw that for Riemann surfaces and tori, holomorphic forms are harmonic and more. For general compact complex manifolds, the relationship is much more complicated. There is, however, an important class of manifolds, called Kähler manifolds, on which these kinds of results do hold. More precisely, harmonic forms on such manifolds decompose into holomorphic, antiholomorphic, and more generally harmonic (p,q) parts. This is the Hodge decomposition, which is the central theorem in the subject.