Holomorphic connections on filtered bundles over curves

Abstract

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