Torsors on moduli spaces of principal $G$-bundles
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911845917392896 |
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| author | Biswas, Indranil Mukhopadhyay, Swarnava |
| author_facet | Biswas, Indranil Mukhopadhyay, Swarnava |
| contents | Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $κ$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_κ$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimesκ) \,=\, 0$, $i\,=\, 1,\, 2$, for all $E_G\,\in\, {\mathcal U}_κ$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_κ$ has a natural splitting, where $U_κ$ is the restriction of $\mathcal{U}_κ$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring $Ω^1_{{M}^{rs}_{G}}$--torsors on the moduli space of regularly stable ${M}^{rs}_{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsors on moduli spaces of principal $G$-bundles Biswas, Indranil Mukhopadhyay, Swarnava Algebraic Geometry Representation Theory 14D20, 14H60, 14D21, 32G34, 17B67 Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $κ$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_κ$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimesκ) \,=\, 0$, $i\,=\, 1,\, 2$, for all $E_G\,\in\, {\mathcal U}_κ$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_κ$ has a natural splitting, where $U_κ$ is the restriction of $\mathcal{U}_κ$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring $Ω^1_{{M}^{rs}_{G}}$--torsors on the moduli space of regularly stable ${M}^{rs}_{G}$. |
| title | Torsors on moduli spaces of principal $G$-bundles |
| topic | Algebraic Geometry Representation Theory 14D20, 14H60, 14D21, 32G34, 17B67 |
| url | https://arxiv.org/abs/2404.12877 |