Meromorphic vector bundles on the Fargues–Fontaine curve

Abstract

We introduce and study the stack of 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 defining and studying an analytification functor that compares the scheme-theoretic 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 meromorphic comparison theorem. We expect it to play a key role in proving that the analytification functor is fully-faithful. Along the way, we give new proofs of what we call the topological and scheme-theoretic 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.