$G$-Connections on principal bundles over complete $G$-varieties
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911768775753728 |
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| author | Khan, Bivas Poddar, Mainak |
| author_facet | Khan, Bivas Poddar, Mainak |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $G$-Connections on principal bundles over complete $G$-varieties Khan, Bivas Poddar, Mainak Algebraic Geometry 14J60, 14M25, 14L30, 53C05 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. |
| title | $G$-Connections on principal bundles over complete $G$-varieties |
| topic | Algebraic Geometry 14J60, 14M25, 14L30, 53C05 |
| url | https://arxiv.org/abs/2402.00131 |