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

Let $C$ be a completely algebraic closed non-archimedean field over ${\mathbb{Q}}_p$ and $\alpha ,r$ be two positive integers. Denote by $B_{\alpha }$ the ring ${\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}(C)\mathrm{/}(\ker \theta {)}^{\alpha }$. This paper first constructs a sheafified $p$-adic Riemann–Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\acute{\mathrm{e}}\mathrm{t}}$, $$R^1{\nu }_{\ast }({\mathrm{G}\mathrm{L}}_r({\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}\mathrm{/}(\ker \theta {)}^{\alpha }))\cong {\mathrm{M}\mathrm{I}\mathrm{C}}_r(X)\{-1\},$$ where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in ${\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}\mathrm{/}(\ker \theta {)}^{\alpha }$, 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 ${\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}\mathrm{/}(\ker \theta {)}^{\alpha }$, and prove that they are small $v$-stacks in the sense of Scholze. These constructions generalize Heuer’s work on $p$-adic Simpson correspondence.