Stable Higgs bundles on compact Gauduchon manifolds

TL;DR

This paper proves that on certain compact complex manifolds with Gauduchon metrics, stable Higgs bundles with trivial tangent bundle must have zero Higgs field, and it shows the Higgs bundle-representation correspondence fails in this setting.

Contribution

It establishes a rigidity result for stable Higgs bundles on Gauduchon manifolds and demonstrates the non-extension of the Higgs bundle-representation correspondence beyond Kähler manifolds.

Findings

Stable SL(r,C)-Higgs bundles with trivial tangent bundle have zero Higgs field.

The Higgs bundle-representation correspondence does not extend to compact Gauduchon manifolds.

Application to quotients of complex semisimple groups illustrates the failure of the correspondence.

Abstract

Let $M$ be a compact complex manifold equipped with a Gauduchon metric. If $TM$ is holomorphically trivial, and (V, \theta) is a stable SL(r,{\mathbb C})-Higgs bundle on $M$, then we show that $\theta= 0$. We show that the correspondence between Higgs bundles and representations of the fundamental group for a compact Kaehler manifold does not extend to compact Gauduchon manifolds. This is done by applying the above result to G/\Gamma, where $\Gamma$ is a discrete torsionfree cocompact subgroup of a complex semisimple group $G$.