A criterion for holomorphic Lie algebroid connections
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912426760339456 |
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| author | Alfaya, David Biswas, Indranil Kumar, Pradip Singh, Anoop |
| author_facet | Alfaya, David Biswas, Indranil Kumar, Pradip Singh, Anoop |
| contents | Given a holomorphic Lie algebroid $(V, ϕ)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V, ϕ)$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V, ϕ)$. If $(V, ϕ)$ is split, then a holomorphic vector bundle $E$ on $X$ admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of $E$ is zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A criterion for holomorphic Lie algebroid connections Alfaya, David Biswas, Indranil Kumar, Pradip Singh, Anoop Algebraic Geometry Differential Geometry 14H60 (Primary) 53D17, 53B15, 32C38 (Secondary) Given a holomorphic Lie algebroid $(V, ϕ)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V, ϕ)$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V, ϕ)$. If $(V, ϕ)$ is split, then a holomorphic vector bundle $E$ on $X$ admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of $E$ is zero. |
| title | A criterion for holomorphic Lie algebroid connections |
| topic | Algebraic Geometry Differential Geometry 14H60 (Primary) 53D17, 53B15, 32C38 (Secondary) |
| url | https://arxiv.org/abs/2506.10514 |