Kobayashi-Hitchin correspondence for twisted vector bundles

TL;DR

This paper establishes a correspondence between stability and Hermite-Einstein metrics for twisted holomorphic vector bundles on compact Kähler manifolds, extending classical results to the twisted setting.

Contribution

It proves the Kobayashi-Hitchin correspondence and its approximate version for twisted holomorphic vector bundles on compact Kähler manifolds.

Findings

Twisted holomorphic vector bundles are $g$-polystable iff $g$-Hermite-Einstein.

Twisted holomorphic vector bundles are $g$-semistable iff approximately $g$-Hermite-Einstein.

Results extend classical Kobayashi-Hitchin correspondence to twisted bundles.

Abstract

We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact K\"ahler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact K\"ahler manifold and $g$ is a K\"ahler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.