A criterion for holomorphic Lie algebroid connections

Abstract

Given a holomorphic Lie algebroid (V,ø) on a compact connected Riemann surface X, we give a necessary and sufficient condition for a holomorphic vector bundle E on X to admit a holomorphic Lie algebroid connection. If (V,ø) is nonsplit, then every holomorphic vector bundle on X admits a holomorphic Lie algebroid connection for (V,ø). If (V,ø) is split, then a holomorphic vector bundle E on X admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of E is zero.