A $p$-adic Simpson correspondence for singular rigid-analytic varieties

TL;DR

This paper establishes a $p$-adic Simpson correspondence for proper rigid-analytic varieties over a non-archimedean field, linking pro-étale vector bundles with Higgs bundles on the $ ext{eh}$-site.

Contribution

It generalizes previous $p$-adic Simpson correspondences to all proper rigid-analytic varieties, expanding the scope of the theory.

Findings

Pro-étale vector bundles are equivalent to Higgs bundles on the $ ext{eh}$-site.

The result extends Faltings and Heuer's work to a broader class of varieties.

Provides a new framework for understanding $p$-adic Hodge theory in rigid-analytic geometry.

Abstract

Let $C$ be a complete, algebraically closed non-archimedean extension of $\mathbb{Q}_p$, and $X$ be a proper rigid-analytic variety over $C$. We show that the category of pro-\'etale vector bundles on $X$ is equivalent to the category of Higgs bundles on the $\eh$-site of $X$, thereby generalizing the work of Faltings and Heuer to arbitrary proper rigid-analytic varieties.