Pure Hodge Structures

Abstract

The Hodge decomposition of the n-th cohomology group of a Kähler manifold is the prototype of a Hodge structure of weight n. In this chapter we study these from a more abstract point of view. In §2.1 and § 2.2 the foundations are laid. Hodge theoretic considerations for various sorts of fundamental classes associated to a subvariety are given in § 2.4. In §2.3 some important concepts are developed which play a central role in the remainder of this book, in particular the concept of a Hodge complex, which is introduced in §2.3. The motivating example comes from the holomorphic De Rham complex on a compact Kähler manifold and is called the Hodge-De Rham complex. However, to show that this indeed gives an example of a Hodge complex follows only after a strong form of the Hodge decomposition is shown to hold.