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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.10369 |
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| _version_ | 1866914681442009088 |
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| author | Grantcharov, Nikolay |
| author_facet | Grantcharov, Nikolay |
| contents | Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle $P\in\text{Bun}_G$ without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, $J_P^{n,PD}(\text{Bun}_G)$, of $\text{Bun}_G$ at $P$. The description is in terms of differential forms on $X^n$ with logarithmic singularities along the diagonals and with coefficients in $(\mathfrak{g}_P^*)^{\boxtimes n}$. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, $\hat{X}^n$, is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of $\text{Bun}_G$ and the $\mathcal{L}ie$ operad. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10369 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic Grantcharov, Nikolay Algebraic Geometry Representation Theory Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle $P\in\text{Bun}_G$ without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, $J_P^{n,PD}(\text{Bun}_G)$, of $\text{Bun}_G$ at $P$. The description is in terms of differential forms on $X^n$ with logarithmic singularities along the diagonals and with coefficients in $(\mathfrak{g}_P^*)^{\boxtimes n}$. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, $\hat{X}^n$, is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of $\text{Bun}_G$ and the $\mathcal{L}ie$ operad. |
| title | Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2402.10369 |