Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces
Indranil Biswas
TL;DR
This work analyzes the existence of equivariant holomorphic Lie algebroid connections on holomorphic principal G-bundles over a compact Riemann surface X with a finite Γ-action. It distinguishes nonsplit and split Γ-equivariant Lie algebroids: for nonsplit V, every Γ-equivariant G-bundle E_G admits an equivariant holomorphic Lie algebroid connection; for split V, existence is equivalent to having an equivariant holomorphic connection, a holomorphic connection, and a universal vanishing-degree condition on line bundles associated to Levi reductions. The authors connect these results to the classical Atiyah framework and extend the discussion to parabolic G-bundles on the quotient Y = X/Γ, yielding parallel criteria there. The findings provide a concrete, obstruction-theoretic criterion (via line bundle degrees) that governs when equivariant Lie algebroid connections exist, and translate naturally to parabolic bundle contexts.
Abstract
For a $Γ$--equivariant holomorphic Lie algebroid $(V,\, φ)$, on a compact Riemann surface $X$ equipped with an action of a finite group $Γ$, 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 $(V,\,φ)$ is nonsplit, then it is proved that every holomorphic principal $G$--bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\,φ)$ 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 $(P,\, L(P),\, χ)$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\, \subset\, G$ and $χ$ is a holomorphic character of $L(P)$, and every $Γ$--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 $χ$ is zero. The correspondence between $Γ$--equivariant principal $G$--bundles over $X$ and parabolic $G$--bundles on $X/Γ$ translates the above result to the context of parabolic $G$--bundles.