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

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.