Lie algebroid connection and Harder-Narasimhan reduction
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arXiv
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| Format: | Preprint |
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2026
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| author | Bansal, Ashima Biswas, Indranil Kumar, Pradip |
| author_facet | Bansal, Ashima Biswas, Indranil Kumar, Pradip |
| contents | Take a holomorphic Lie algebroid $(V,\, ϕ)$ on a compact connected Riemann surface $X$ such that the anchor map $ϕ$ is not surjective. Let $P$ be a parabolic subgroup of a complex reductive affine algebraic group $G$ and $E_P\, \subset\, E_G$ a holomorphic reduction of structure group, to $P$, of a holomorphic principal $G$--bundle $E_G$ on $X$. We prove that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,ϕ)$ if the reduction $E_P$ is infinitesimally rigid. If $E_P$ is the Harder--Narasimhan reduction of $E_G$, then it is shown that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,ϕ)$. In particular, for any point $x_0\,\in\, X$, the Harder--Narasimhan reduction $E_P$ admits a logarithmic connection that is nonsingular on the complement $X\setminus\{x_0\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lie algebroid connection and Harder-Narasimhan reduction Bansal, Ashima Biswas, Indranil Kumar, Pradip Algebraic Geometry 14H60, 53D17, 53B15, 32C38 Take a holomorphic Lie algebroid $(V,\, ϕ)$ on a compact connected Riemann surface $X$ such that the anchor map $ϕ$ is not surjective. Let $P$ be a parabolic subgroup of a complex reductive affine algebraic group $G$ and $E_P\, \subset\, E_G$ a holomorphic reduction of structure group, to $P$, of a holomorphic principal $G$--bundle $E_G$ on $X$. We prove that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,ϕ)$ if the reduction $E_P$ is infinitesimally rigid. If $E_P$ is the Harder--Narasimhan reduction of $E_G$, then it is shown that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,ϕ)$. In particular, for any point $x_0\,\in\, X$, the Harder--Narasimhan reduction $E_P$ admits a logarithmic connection that is nonsingular on the complement $X\setminus\{x_0\}$. |
| title | Lie algebroid connection and Harder-Narasimhan reduction |
| topic | Algebraic Geometry 14H60, 53D17, 53B15, 32C38 |
| url | https://arxiv.org/abs/2601.18169 |