Holomorphic Lie Algebroid Connections on Holomorphic Principal Bundles on Compact Riemann Surfaces
Vol. 33 (2026), No. 1, Page 97-125.
Biswas, I.
Holomorphic Lie Algebroid Connections on Holomorphic Principal Bundles on Compact Riemann Surfaces
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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:
(1) An equivariant principal $G-$bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection.
(2) The equivariant principal $G-$bundle $E_G$ admits an equivariant holomorphic connection.
(3) The principal $G-$bundle $E_G$ admits a holomorphic connection.
(4) 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.
Keywords: Lie algebroid, holomorphic connection, principal bundle, Atiyah bundle, reductive group.
Mathematics Subject Classification (2020): 14H60, 53D17, 53B15, 32C38.
Received: 2025-07-08
