Hodge classes of Chern character forms on compact K\'ahler manifolds

In this paper we show that every rational cohomology class of type $(p,p)$ on a compact K\"ahler manifold can be representated as a differential $(p,p)$-form given by an explicit formula involving a \v{C}ech cocycle. First we represent Chern characters of smooth vector bundles by \v{C}ech 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 K\"ahler.