Einstein-Hermitian connection on twisted Higgs bundles

TL;DR

This paper proves that twisted Higgs bundles on smooth projective varieties admit Einstein-Hermitian connections if and only if they are polystable, simplifying previous proofs for the non-Higgs case.

Contribution

It provides a simpler proof of the polystability and Einstein-Hermitian connection correspondence for twisted Higgs bundles.

Findings

Twisted Higgs bundles are polystable iff they admit Einstein-Hermitian connections.

The proof approach is simpler than previous methods.

Extends known results from twisted vector bundles to Higgs bundles.

Abstract

Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,\theta)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,\theta)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.