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

TL;DR

This paper provides an explicit formula for representing rational cohomology classes of type (p,p) on compact Kähler manifolds as differential forms, linking Chern characters to the Hodge structure via Čech cocycles.

Contribution

It introduces a new explicit formula for representing (p,p) cohomology classes using Čech cocycles and analyzes their behavior under the Hodge structure on compact Kähler manifolds.

Findings

Explicit formula for (p,p) classes as differential forms

Representation of Chern characters via Čech cocycles

Analysis of cocycle behavior with respect to Hodge structure

Abstract

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.