Coupled connections on a compact Riemann surface

Abstract

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K^⊗n_X⊗E, where K_X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.