Existence of holomorphic Lie algebroid connections in higher dimensions

TL;DR

This paper provides a necessary and sufficient condition for the existence of holomorphic Lie algebroid connections on vector bundles over higher-dimensional complex projective varieties.

Contribution

It extends the theory of holomorphic Lie algebroid connections by characterizing their existence in higher dimensions.

Findings

Established a criterion for the existence of holomorphic $(V, )$-connections.

Applied the criterion to complex projective varieties of dimension at least three.

Contributed to the understanding of holomorphic Lie algebroid structures in higher-dimensional geometry.

Abstract

Let $(V, \phi)$ be a holomorphic Lie algebroid over an irreducible smooth complex projective variety $X$ of dimension at least three, and let $E$ be a holomorphic vector bundle on $X$. We establish a necessary and sufficient condition for the existence of a holomorphic $(V, \phi)$--connection on $E$.