Torsors over moduli spaces of vector bundles over curves of fixed determinant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915376943595520 |
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| author | Biswas, Indranil Hurtubise, Jacques |
| author_facet | Biswas, Indranil Hurtubise, Jacques |
| contents | Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $ξ$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $χ(E\otimes F)= 0$ for $E \in \mathcal M$. Then any $E\in \mathcal M$ with $H^0(X, E\otimes F) = 0 = H^1(X, E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\, \nabla)$, where $E\, \in\, \mathcal M$ and $\nabla$ is a holomorphic projective connection on $E$, is an algebraic $T^*{\mathcal M}$--torsor on $\mathcal M$. We identify this $T^*{\mathcal M}$--torsor on $\mathcal M$ with the $T^*{\mathcal M}$--torsor given by the sheaf of connections on an ample line bundle over $\mathcal M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torsors over moduli spaces of vector bundles over curves of fixed determinant Biswas, Indranil Hurtubise, Jacques Algebraic Geometry Differential Geometry Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $ξ$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $χ(E\otimes F)= 0$ for $E \in \mathcal M$. Then any $E\in \mathcal M$ with $H^0(X, E\otimes F) = 0 = H^1(X, E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\, \nabla)$, where $E\, \in\, \mathcal M$ and $\nabla$ is a holomorphic projective connection on $E$, is an algebraic $T^*{\mathcal M}$--torsor on $\mathcal M$. We identify this $T^*{\mathcal M}$--torsor on $\mathcal M$ with the $T^*{\mathcal M}$--torsor given by the sheaf of connections on an ample line bundle over $\mathcal M$. |
| title | Torsors over moduli spaces of vector bundles over curves of fixed determinant |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2507.05690 |