Lie algebroid connection and 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 $X\setminus\{x_0\}$.