Meromorphic vector bundles on the Fargues--Fontaine curve

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 ${\mathrm{Bun}}_G$. 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 ${\mathrm{Bun}}_G^{\mathrm{mer}}$ 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 ${\mathrm{Bun}}_G$ and $\mathfrak{B}(G)$ are reversed and that the two stacks take the same values when evaluated on schemes.

Final version, to appear in J. Eur. Math. Soc