A criterion for holomorphic Lie algebroid connections

Abstract

Given a holomorphic Lie algebroid $(V,\varphi )$ 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 $(V,\varphi )$ is nonsplit, then every holomorphic vector bundle on X admits a holomorphic Lie algebroid connection for $(V,\varphi )$. If $(V,\varphi )$ 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.