Parabolic vector bundles and Lie algebroid connections

TL;DR

This paper develops the theory of parabolic Lie algebroid connections on vector bundles over Riemann surfaces, extending classical concepts and providing a full characterization of bundles admitting such connections.

Contribution

It introduces parabolic Lie algebroid connections, constructs an Atiyah sequence analogue, and characterizes bundles with these connections for stable underlying bundles.

Findings

Defined parabolic Lie algebroid connections on parabolic vector bundles

Constructed an analogue of the Atiyah exact sequence for parabolic Lie algebroids

Provided a complete characterization of bundles admitting these connections for stable cases

Abstract

Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.