Hodge classes of Chern character forms on compact Kähler manifolds

Abstract

In this paper we show that every rational cohomology class of type (p, p) on a compact Kähler manifold can be representated as a differential (p, p)-form given by an explicit formula involving a Čech cocycle. First we represent Chern characters of smooth vector bundles by Čech cocycles with values in the sheaf of differential forms. We then consider the behavior of these cocycles with respect to the Hodge structure on cohomology when the base manifold is compact Kähler.