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.
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