A criterion for Lie algebroid connections on a compact Riemann surface

TL;DR

This paper establishes a precise criterion for when holomorphic vector bundles on a compact Riemann surface admit Lie algebroid connections, focusing on the stability of the underlying Lie algebroid.

Contribution

It provides a necessary and sufficient condition for the existence of Lie algebroid connections on stable holomorphic vector bundles over Riemann surfaces.

Findings

Derived a criterion linking bundle stability to Lie algebroid connection existence

Characterized Lie algebroid connections in terms of bundle properties

Focused on compact Riemann surfaces with stable Lie algebroids

Abstract

Let $X$ be a compact connected Riemann surface and $(V, \phi)$ a holomorphic Lie algebroid on $X$ such that the holomorphic vector bundle $V$ is stable. We give a necessary and sufficient condition on holomorphic vector bundles $E$ on $X$ to admit a Lie algebroid connection.