A criterion for parabolic vector bundles to admit a parabolic Lie algebroid connection

TL;DR

This paper establishes a precise criterion for when a parabolic vector bundle on a Riemann surface admits a compatible parabolic Lie algebroid connection, linking geometric structures.

Contribution

It provides a necessary and sufficient condition for the existence of a parabolic Lie algebroid connection on a parabolic vector bundle over a Riemann surface.

Findings

Derived a criterion for parabolic Lie algebroid connections

Connected the existence of connections to geometric conditions

Focused on bundles over compact Riemann surfaces

Abstract

Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a parabolic vector bundle on $X$, with parabolic structure over a nonzero reduced effective divisor, to admit a parabolic Lie algebroid connection for the Lie algebroid $(V, \phi)$.