Lie algebroid connection and Harder-Narasimhan reduction
Ashima Bansal, Indranil Biswas, Pradip Kumar
TL;DR
This work proves that a holomorphic reduction $E_P$ of a principal $G$-bundle to a parabolic subgroup admits a holomorphic Lie algebroid connection for a given holomorphic Lie algebroid $(V,\phi)$ on a compact Riemann surface provided the anchor is not surjective and the reduction is infinitesimally rigid, i.e. $H^0(X,\mathrm{ad}(E_G)/\mathrm{ad}(E_P))=0$. The central mechanism is a splitting argument for an exact sequence associated with the Lie algebroid connection, leveraging the Levi quotient and the vanishing of the obstruction class via Serre duality. As a corollary, when $E_P$ is the Harder--Narasimhan reduction, one obtains a logarithmic connection nonsingular on $X\setminus\{x_0\}$ for any $x_0$, and more generally, a filtration-preserving connection for the Harder--Narasimhan filtration of a vector bundle. The results also show that unstable reductions cannot admit a holomorphic connection, and they extend to yield logarithmic and Lie algebroid connections that preserve filtrations in the corresponding vector bundle setting.
Abstract
Take a holomorphic Lie algebroid $(V,\, φ)$ on a compact connected Riemann surface $X$ such that the anchor map $φ$ 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 $(V,\,φ)$ 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 $(V,\,φ)$. 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 $X\setminus\{x_0\}$.