$G$-Connections on principal bundles over complete $G$-varieties

Bivas Khan; Mainak Poddar

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arXiv · 2402.00131

$G$-Connections on principal bundles over complete $G$-varieties

Bivas Khan·Mainak Poddar

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Abstract

Let $X$ be a complete variety over an algebraically closed field $k$ of characteristic zero, equipped with an action of an algebraic group $G$. Let $H$ be a reductive group. We study the notion of $G$-connection on a principal $H$-bundle. We give necessary and sufficient criteria for the existence of $G$-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of $G$-connection and equivariant structure on a principal $H$-bundle, under the assumption that $G$ is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.

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Bivas Khan, Mainak Poddar. 2024-01-31. $G$-Connections on principal bundles over complete $G$-varieties. https://arxiv.org/abs/2402.00131

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