A criterion for holomorphic Lie algebroid connections

TL;DR

This paper establishes a precise criterion for when holomorphic vector bundles on a Riemann surface admit connections compatible with a given holomorphic Lie algebroid, depending on the algebroid's splitting properties.

Contribution

It provides a necessary and sufficient condition for the existence of holomorphic Lie algebroid connections on vector bundles over Riemann surfaces, clarifying the role of the algebroid's splitting.

Findings

Non-split algebroids allow all bundles to admit connections.

Split algebroids require bundles to have zero degree components.

The criterion depends on the splitting property of the Lie algebroid.

Abstract

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