Lie algebroid connection and Harder-Narasimhan reduction

TL;DR

This paper investigates conditions under which holomorphic reductions of principal bundles on Riemann surfaces admit Lie algebroid connections, especially focusing on the Harder-Narasimhan reduction and its implications for logarithmic connections.

Contribution

It establishes that infinitesimally rigid reductions and Harder-Narasimhan reductions admit holomorphic Lie algebroid connections, extending the understanding of connections on principal bundles.

Findings

Harder-Narasimhan reduction admits a Lie algebroid connection.

Infinitesimally rigid reductions admit a Lie algebroid connection.

Existence of logarithmic connections on the Harder-Narasimhan reduction.

Abstract

Take a holomorphic Lie algebroid $(V,\, \phi)$ on a compact connected Riemann surface $X$ such that the anchor map $\phi$ is not surjective. Let $P$ be a parabolic subgroup of a complex reductive affine algebraic group $G$ and $E_P\, \subset\, E_G$ a holomorphic reduction of structure group, to $P$, of a holomorphic principal $G$--bundle $E_G$ on $X$. We prove that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,\phi)$ if the reduction $E_P$ is infinitesimally rigid. If $E_P$ is the Harder--Narasimhan reduction of $E_G$, then it is shown that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,\phi)$. In particular, for any point $x_0\,\in\, X$, the Harder--Narasimhan reduction $E_P$ admits a logarithmic connection that is nonsingular on the complement $X\setminus\{x_0\}$.