Meromorphic vector bundles on the Fargues--Fontaine curve

Author:

Ian Gleason, Alexander B. Ivanov

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-07-02 16:00:00

Abstract

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.