Representability of Chern-Weil forms

TL;DR

This paper investigates conditions under which Chern-Weil forms can be represented by specific metrics, providing evidence for Griffiths' conjecture in certain vector bundle cases and exploring a rank-2 bundle related to vortex equations.

Contribution

It offers new insights into the representability of Chern-Weil forms, including evidence supporting Griffiths' conjecture and analysis of a special rank-2 bundle over a Riemann surface product.

Findings

Evidence for Griffiths' conjecture in semi-stable Hartshorne-ample vector bundles.

Construction of metrics with positive Chern forms in specific cases.

Analysis of a rank-2 bundle related to vortex equations.

Abstract

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.