Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces

TL;DR

This paper studies equivariant holomorphic Lie algebroid connections on principal bundles over compact Riemann surfaces, establishing existence results and equivalences related to holomorphic and equivariant connections, with implications for parabolic bundles.

Contribution

It proves existence of equivariant holomorphic Lie algebroid connections for nonsplit cases and characterizes when such connections exist in split cases, linking to classical holomorphic connections and degrees of associated line bundles.

Findings

Existence of equivariant holomorphic Lie algebroid connections for nonsplit cases.

Equivalence of conditions for the existence of equivariant holomorphic Lie algebroid and holomorphic connections in split cases.

Correspondence between equivariant principal bundles and parabolic bundles on quotients.

Abstract

For a $\Gamma$--equivariant holomorphic Lie algebroid $(V,\, \phi)$, on a compact Riemann surface $X$ equipped with an action of a finite group $\Gamma$, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal $G$--bundles over $X$, where $G$ is a connected affine complex reductive group. If $(V,\,\phi)$ is nonsplit, then it is proved that every holomorphic principal $G$--bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\,\phi)$ is split, then it is proved that the following four statements are equivalent: An equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection. The equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic connection. The principal $G$--bundle $E_G$ admits a holomorphic connection. For every triple $(P,\, L(P),\, \chi)$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\, \subset\, G$ and $\chi$ is a holomorphic character of $L(P)$, and every $\Gamma$--equivariant holomorphic reduction of structure group $E_{L(P)}$ of $E_G$ to $L(P)$, the degree of the line bundle over $X$ associated to $E_{L(P)}$ for $\chi$ is zero. The correspondence between $\Gamma$--equivariant principal $G$--bundles over $X$ and parabolic $G$--bundles on $X/\Gamma$ translates the above result to the context of parabolic $G$--bundles.