Meromorphic vector bundles on the Fargues--Fontaine curve
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910222450163712 |
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| author | Gleason, Ian Ivanov, Alexander B. Zillinger, Felix |
| author_facet | Gleason, Ian Ivanov, Alexander B. Zillinger, Felix |
| contents | We introduce and study the stack of \textit{meromorphic} $G$-bundles on the Fargues--Fontaine curve. This object defines a correspondence between the Kottwitz stack $\mathfrak{B}(G)$ and $\operatorname{Bun}_G$. We expect it to play a crucial role in comparing the schematic and analytic versions of the geometric local Langlands categories. Our first main result is the identification of the generic Newton strata of ${\operatorname{Bun}}_G^{\operatorname{mer}}$ with the Fargues--Scholze charts $\mathcal{M}$. Our second main result is a generalization of Fargues' theorem in families. We call this the \textit{meromorphic comparison theorem}. It plays a key role in proving that the analytification functor is fully faithful. Along the way, we give new proofs to what we call the \textit{topological and schematic comparison theorems}. These say that the topologies of $\operatorname{Bun}_G$ and $\mathfrak{B}(G)$ are reversed and that the two stacks take the same values when evaluated on schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Meromorphic vector bundles on the Fargues--Fontaine curve Gleason, Ian Ivanov, Alexander B. Zillinger, Felix Algebraic Geometry Number Theory We introduce and study the stack of \textit{meromorphic} $G$-bundles on the Fargues--Fontaine curve. This object defines a correspondence between the Kottwitz stack $\mathfrak{B}(G)$ and $\operatorname{Bun}_G$. We expect it to play a crucial role in comparing the schematic and analytic versions of the geometric local Langlands categories. Our first main result is the identification of the generic Newton strata of ${\operatorname{Bun}}_G^{\operatorname{mer}}$ with the Fargues--Scholze charts $\mathcal{M}$. Our second main result is a generalization of Fargues' theorem in families. We call this the \textit{meromorphic comparison theorem}. It plays a key role in proving that the analytification functor is fully faithful. Along the way, we give new proofs to what we call the \textit{topological and schematic comparison theorems}. These say that the topologies of $\operatorname{Bun}_G$ and $\mathfrak{B}(G)$ are reversed and that the two stacks take the same values when evaluated on schemes. |
| title | Meromorphic vector bundles on the Fargues--Fontaine curve |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2307.00887 |