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A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

Yudong Liu, Chenglong Ma, Xiecheng Nie, Xiaoyu Qu, Yupeng Wang

TL;DR

This work develops a stacky $p$-adic Riemann--Hilbert correspondence for Hitchin-small objects on a semi-stable formal scheme over a perfectoid base ${\mathcal O}_C$, through the construction of a period sheaf with connection ${\mathcal O}{\bf B}_{\rm dR,{\rm pd}}^+,d$ on the $v$-site. The authors prove a moduli-level equivalence between Hitchin-small ${\bf B}_{\rm dR,n}^+$-local systems on $X_v$ and Hitchin-small integrable connections on a fixed lifting ${\widetilde X}_n$ of the generic fiber $X$, for all $n\ge 1$, and they establish this via a globally defined period sheaf satisfying a stacky Riemann--Hilbert correspondence. The strategy hinges on local-to-global arguments: a local small Riemann--Hilbert theory (via a local Simpson correspondence for a Gamma-action) combined with exactification of log-structures and chart-independent constructions of the period sheaf, then assembled into a stack-level equivalence. The results extend the non-abelian $p$-adic Hodge theory program to the Hitchin-small locus over a perfectoid base and yield applications to arithmetic RH in good reduction, as well as to moduli problems in $p$-adic geometry and Higgs/bundle correspondences. The framework provides a robust bridge between ${\bf B}_{\rm dR}^+$-local systems and Higgs/flat-connection data in a stack-theoretic setting, with potential further extensions to broader base fields and moduli spaces.

Abstract

Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.

A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

TL;DR

This work develops a stacky -adic Riemann--Hilbert correspondence for Hitchin-small objects on a semi-stable formal scheme over a perfectoid base , through the construction of a period sheaf with connection on the -site. The authors prove a moduli-level equivalence between Hitchin-small -local systems on and Hitchin-small integrable connections on a fixed lifting of the generic fiber , for all , and they establish this via a globally defined period sheaf satisfying a stacky Riemann--Hilbert correspondence. The strategy hinges on local-to-global arguments: a local small Riemann--Hilbert theory (via a local Simpson correspondence for a Gamma-action) combined with exactification of log-structures and chart-independent constructions of the period sheaf, then assembled into a stack-level equivalence. The results extend the non-abelian -adic Hodge theory program to the Hitchin-small locus over a perfectoid base and yield applications to arithmetic RH in good reduction, as well as to moduli problems in -adic geometry and Higgs/bundle correspondences. The framework provides a robust bridge between -local systems and Higgs/flat-connection data in a stack-theoretic setting, with potential further extensions to broader base fields and moduli spaces.

Abstract

Let be an algebraically closed perfectoid field over with the ring of integer and the infinitesimal thickening . Let be a semi-stable formal scheme over with a fixed flat lifting over . Let be the generic fiber of and be its lifting over induced by . Let and be the -stacks of rank- Hitchin-small integrable connections on and -local systems on , respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection on .
Paper Structure (14 sections, 33 theorems, 309 equations)

This paper contains 14 sections, 33 theorems, 309 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Strategy for the proof and organization of this paper
  5. Notations
  6. Review of semi-stable formal schemes
  7. The period sheaf with connection $({\mathcal{O}}{\mathbb B}_{{\mathrm{dR}},{\mathrm{pd}}}^+,{\mathrm d})$
  8. Construction of $({\mathcal{O}}{\mathbb B}_{{\mathrm{dR}},{\mathrm{pd}}}^+,{\mathrm d})$: The small affine case
  9. Construction of $({\mathcal{O}}{\mathbb B}_{{\mathrm{dR}},{\mathrm{pd}}}^+,{\mathrm d})$: The general case
  10. Local small Riemann--Hilbert correspondence
  11. A criterion for being Hitchin-small
  12. Local small Simpson correspondence
  13. Local small Riemann--Hilbert correspondence
  14. The stacky Riemann--Hilbert correspondence: Proof of Theorem \ref{['thm:Stacky RH']}

Key Result

Theorem 1.1

For any $n\geq 1$, the flat lifting $\widetilde{{\mathfrak X}}$ of ${\mathfrak X}$ induces an equivalence between the moduli stack ${\mathrm L}{\mathrm S}_r(X,{{\mathbb B}_{\mathrm{dR},n}^+})^{{{\rm H}\text{-sm}}}$ of Hitchin-small ${{\mathbb B}_{\mathrm{dR},n}^+}$-local systems of rank $r$ on $X_{v}$ and the moduli stack ${\mathrm{MIC}}_r(\widetilde{X}_n)^{{\rm H}\text{-sm}}$ of Hitchin-small in

Theorems & Definitions (87)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • proof
  • Remark 1.8
  • Remark 1.9
  • ...and 77 more