A criterion for the existence of a flat connection on a parabolic vector bundle

Abstract

We define holomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a holomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the para- bolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic connection if and only if every direct summand of it is of degree zero. 2000 Mathematics Subject Classification. 14H60, 32L05