Principle bundles admitting a holomorphic structure

Abstract

Let $M$ be a compact connected K\"ahler manifold and let ${\E}_{l-1}$ be the smallest term in the Harder-Narasimhan filtration of its tangent bundle. Let $G$ be an affine algebraic reductive group over $\C$. We prove the following result: If $M$ satisfies the condition that $\deg (T/{\E}_{l-1}) \geq 0$, then a holomorphic principal $G$-bundle $P$ on $M$ admitting a compatible holomorphic connection is semistable. Moreover, if $\deg (T/{\E}_{l-1}) >0$, then such a bundle $P$ actually admits a compatible flat $G$-connection.