Holomorphic connections on filtered bundles over curves

Abstract

Let $X$ be a compact connected Riemann surface and $E_P$ a holomorphic principal $P$--bundle over $X$, where $P$ is a parabolic subgroup of a complex reductive affine algebraic group $G$. If the Levi bundle associated to $E_P$ admits a holomorphic connection, and the reduction $E_P \subset E_P\times^P G$ is rigid, we prove that $E_P$ admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition for a filtered holomorphic vector bundle over $X$ to admit a filtration preserving holomorphic connection. Moreover, we state a weaker sufficient condition in the special case of a filtration of length two.