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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2412.19116 |
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| _version_ | 1866909441455030272 |
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| author | Jakob, Konstantin Yun, Zhiwei |
| author_facet | Jakob, Konstantin Yun, Zhiwei |
| contents | For a reductive group $G$ over a finite field $k$, and a smooth projective curve $X/k$, we give a motivic counting formula for the number of absolutely indecomposable $G$-bundles on $X$. We prove that the counting can be expressed via the cohomology of the moduli stack of stable parabolic $G$-Higgs bundles on $X$. This result generalizes work of Schiffmann and work of Dobrovolska, Ginzburg, and Travkin from $\mathrm{GL}_n$ to a general reductive group. Along the way we prove some structural results on automorphism groups of $G$-torsors, and we study certain Lie-theoretic counting problems related to the case when $X$ is an elliptic curve - a case which we investigate more carefully following Fratila, Gunningham and P. Li. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting absolutely indecomposable $G$-bundles Jakob, Konstantin Yun, Zhiwei Algebraic Geometry For a reductive group $G$ over a finite field $k$, and a smooth projective curve $X/k$, we give a motivic counting formula for the number of absolutely indecomposable $G$-bundles on $X$. We prove that the counting can be expressed via the cohomology of the moduli stack of stable parabolic $G$-Higgs bundles on $X$. This result generalizes work of Schiffmann and work of Dobrovolska, Ginzburg, and Travkin from $\mathrm{GL}_n$ to a general reductive group. Along the way we prove some structural results on automorphism groups of $G$-torsors, and we study certain Lie-theoretic counting problems related to the case when $X$ is an elliptic curve - a case which we investigate more carefully following Fratila, Gunningham and P. Li. |
| title | Counting absolutely indecomposable $G$-bundles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2412.19116 |