Torelli theorem for moduli stacks of vector bundles and principal G-bundles
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929602742452224 |
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| author | Alfaya, David Biswas, Indranil Gómez, Tomás L. Mukhopadhyay, Swarnava |
| author_facet | Alfaya, David Biswas, Indranil Gómez, Tomás L. Mukhopadhyay, Swarnava |
| contents | Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve).
We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07666 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Torelli theorem for moduli stacks of vector bundles and principal G-bundles Alfaya, David Biswas, Indranil Gómez, Tomás L. Mukhopadhyay, Swarnava Algebraic Geometry 14C34 (Primary) 14H60, 14D23 (Secondary) Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve). We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group. |
| title | Torelli theorem for moduli stacks of vector bundles and principal G-bundles |
| topic | Algebraic Geometry 14C34 (Primary) 14H60, 14D23 (Secondary) |
| url | https://arxiv.org/abs/2310.07666 |