Meromorphic vector bundles on the Fargues--Fontaine curve

TL;DR

This paper introduces the stack of meromorphic G-bundles on the Fargues--Fontaine curve, establishing key correspondences and generalizations that advance the understanding of geometric local Langlands categories.

Contribution

It defines the meromorphic G-bundles stack, identifies generic Newton strata with Fargues--Scholze charts, and generalizes Fargues' theorem in families, enhancing the framework for geometric Langlands.

Findings

Identified generic Newton strata with Fargues--Scholze charts.

Proved the meromorphic comparison theorem.

Provided new proofs of topological and schematic comparison theorems.

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.