On smooth adic spaces over ${\mathbb{B}}_{\mathrm{dR}}^{+}$ and sheafified $p$-adic Riemann--Hilbert correspondence

Abstract

Let $C$ be a completely algebraic closed non-archimedean field over ${\mathbb{Q}}_p$ and $\hat{I}\pm ,r$ be two positive integers. Denote by $B_{\hat{I}}\pm$ the ring ${\mathbb{B}}_{\mathrm{dR}}^{+}(C)/(\ker \hat{I}¸{)}^{\hat{I}}\pm$. This paper first constructs a sheafified $p$-adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\acute{\mathrm{e}}\mathrm{t}}$, \[ R^1ν_*\big( \mathrm{GL}_r(\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α) \big) \cong \mathrm{MIC}_{r}(X)\{-1\}, \] where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in ${\mathbb{B}}_{\mathrm{dR}}^{+}/(\ker \hat{I}¸{)}^{\hat{I}}\pm$, and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank $r$. We then define the moduli space of integrable connections on $X$ and the moduli space of $v$-vector bundles on $X$ with coefficients in ${\mathbb{B}}_{\mathrm{dR}}^{+}/(\ker \hat{I}¸{)}^{\hat{I}}\pm$, and prove that they are small $v$-stacks in the sense of Scholze. These constructions generalize Heuer's work on $p$-adic Simpson correspondence.