Stable Higgs bundles and Hermitian-Einstein metrics on non-K\"ahler manifolds

TL;DR

This paper proves that on compact Gauduchon manifolds, a modified Donaldson heat flow converges to solutions of a generalized Hermitian-Einstein equation for stable Higgs bundles, extending classical results to non-Kähler settings.

Contribution

It introduces a convergence result for a modified Donaldson heat flow on non-Kähler manifolds, establishing existence of Hermitian-Einstein metrics for stable Higgs bundles.

Findings

Convergence of the modified heat flow to Hermitian-Einstein metrics.

Extension of stability and Hermitian-Einstein correspondence to non-Kähler manifolds.

Validation of the generalized Hermitian-Einstein equation in this setting.

Abstract

Let $X$ be a compact Gauduchon manifold, and let $E$ and $V_0$ be holomorphic vector bundles over $X$. Suppose that $E$ is stable when considering all subsheaves preserved by a Higgs field $\theta\in H^0($End$(E)\otimes V_0)$. Then a modified version of the Donaldson heat flow converges along a subsequence of times to a solution of a generalized Hermitian-Einstein equation, given by $i\Lambda F+[\theta,\theta^\dagger]=\lambda I$.