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Moduli spaces in $p$-adic non-abelian Hodge theory

Ben Heuer

Abstract

We propose a new moduli-theoretic approach to the $p$-adic Simpson correspondence for a smooth proper rigid space $X$ over $\mathbb C_p$ with coefficients in any rigid analytic group $G$, in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism \[R^1ν_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where $ν:Y_{v}\to Y_{et}$ is the natural morphism of sites, and where $\mathrm{Higgs}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on $Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson correspondence to $G$-bundles and to perfectoid families. We apply these results to deduce $v$-descent criteria for étale $G$-bundles which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathscr Higgs_G$. As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism $\mathscr Bun_{G,v}\to \mathcal A_G$ from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin base. This allows us to give a conjectural reformulation of the $p$-adic Simpson correspondence for $X$ in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms.

Moduli spaces in $p$-adic non-abelian Hodge theory

Abstract

We propose a new moduli-theoretic approach to the -adic Simpson correspondence for a smooth proper rigid space over with coefficients in any rigid analytic group , in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative -adic Hodge theory. For any smoothoid space , we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism where is the natural morphism of sites, and where is the sheaf of isomorphism classes of -Higgs bundles on . We also prove a generalisation of Faltings' local -adic Simpson correspondence to -bundles and to perfectoid families. We apply these results to deduce -descent criteria for étale -bundles which show that -Higgs bundles on form a small -stack . As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism from the small -stack of -topological -bundles on to the Hitchin base. This allows us to give a conjectural reformulation of the -adic Simpson correspondence for in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms.
Paper Structure (52 sections, 83 theorems, 174 equations)

This paper contains 52 sections, 83 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. The sheafified non-abelian Hodge correspondence
  3. Moduli spaces in $p$-adic non-abelian Hodge theory
  4. The Hitchin morphism on the Betti side
  5. Towards a $p$-adic Corlette--Simpson correspondence for $\pi^{{\operatorname{\acute{e}t}}}_1(X)$
  6. Application: $G$-torsors on rigid spaces in the $v$-topology
  7. Relation to Faltings' global $p$-adic Simpson correspondence
  8. Relation to other previous works, and outlook
  9. Smoothoid spaces
  10. Definition of smoothoid spaces and toric charts
  11. Differentials on smoothoid spaces via Hodge--Tate comparison
  12. Computations with toric charts
  13. Non-cyclotomic base fields
  14. Higgs bundles on smoothoid spaces
  15. Rigid groups as $v$-sheaves
  16. ...and 37 more sections

Key Result

Theorem 1.2

Let $X$ be any smooth rigid space over $K$ and let $\nu:X_{v}\to X_{{\operatorname{\acute{e}t}}}$ be the natural morphism of sites. Let $G$ be a rigid group over $K$, considered as a sheaf on $X_{v}$. For example, $G$ could be the analytification of any algebraic group. Then there is a canonical iso of sheaves of pointed sets on $X_{{\operatorname{\acute{e}t}}}$ that is functorial in $X$, $G$ and

Theorems & Definitions (199)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7: HX
  • Theorem 1.8: heuer-diamantine-Picard
  • Corollary 1.9
  • Theorem 1.10
  • ...and 189 more