Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds

TL;DR

This paper establishes the equivalence between semi-stability and the existence of approximate Hermitian--Einstein structures for twisted holomorphic vector bundles over compact Gauduchon manifolds, extending classical results and inequalities.

Contribution

It proves the equivalence of semi-stability and approximate Hermitian--Einstein structures for twisted bundles on Gauduchon manifolds, and extends the Bogomolov inequality to this setting.

Findings

Semi-stability is equivalent to approximate Hermitian--Einstein structures.

The Bogomolov type inequality holds for semi-stable twisted bundles.

Application of Uhlenbeck--Yau's continuity method to Gauduchon manifolds.

Abstract

In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck--Yau's continuity method, we show that the existence of approximate Hermitian--Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.