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A $p$-adic Simpson correspondence for smooth proper rigid varieties

Ben Heuer

TL;DR

The paper proves a global $p$-adic Simpson correspondence for smooth proper rigid varieties over a complete algebraically closed $p$-adic field, establishing an exact tensor equivalence between pro-étale vector bundles and Higgs bundles, conditioned on a lift to $B_{ ext{dR}}^+/\xi^2$ and a $p$-adic exponential. The approach uses spectral covers and moduli spaces of invertible modules on pro-étale sites, leveraging a multiplicative Hodge–Tate sequence and Higgs–Tate torsors to twist and exponentiate bundles, yielding a canonical, choice-dependent but natural equivalence. A local-to-global strategy via the Local correspondence (and Rodríguez Camargo’s Higgs field) couples toric analyses with global twisting to produce the global functor and its inverse, compatible with tensor products. The authors also prove a cohomological comparison $R u_*(V)\, ilde{=}\, ext{Higgs complex}$, yielding $Roldsymbol m{H}^*(X_{ ext{pro-ét}},V) \,\cong\, Roldsymbol m{H}^*(X,(E, heta))$, thereby extending p-adic Hodge-type decompositions to a non-abelian, higher-dimensional setting and general rigid-analytic contexts. This advances $p$-adic non-abelian Hodge theory beyond curves, with strong links to spectral geometry and deformation-theoretic perspectives on the Simpson correspondence."

Abstract

For any smooth proper rigid analytic space $X$ over a complete algebraically closed extension of $\mathbb Q_p$, we construct a $p$-adic Simpson correspondence: an equivalence of categories between vector bundles on Scholze's pro-étale site of $X$ and Higgs bundles on $X$. This generalises a result of Faltings from smooth projective curves to any higher dimension, and further to the rigid analytic setup. The strategy is new, and is based on the study of rigid analytic moduli spaces of pro-étale invertible sheaves on spectral varieties.

A $p$-adic Simpson correspondence for smooth proper rigid varieties

TL;DR

The paper proves a global -adic Simpson correspondence for smooth proper rigid varieties over a complete algebraically closed -adic field, establishing an exact tensor equivalence between pro-étale vector bundles and Higgs bundles, conditioned on a lift to and a -adic exponential. The approach uses spectral covers and moduli spaces of invertible modules on pro-étale sites, leveraging a multiplicative Hodge–Tate sequence and Higgs–Tate torsors to twist and exponentiate bundles, yielding a canonical, choice-dependent but natural equivalence. A local-to-global strategy via the Local correspondence (and Rodríguez Camargo’s Higgs field) couples toric analyses with global twisting to produce the global functor and its inverse, compatible with tensor products. The authors also prove a cohomological comparison , yielding , thereby extending p-adic Hodge-type decompositions to a non-abelian, higher-dimensional setting and general rigid-analytic contexts. This advances -adic non-abelian Hodge theory beyond curves, with strong links to spectral geometry and deformation-theoretic perspectives on the Simpson correspondence."

Abstract

For any smooth proper rigid analytic space over a complete algebraically closed extension of , we construct a -adic Simpson correspondence: an equivalence of categories between vector bundles on Scholze's pro-étale site of and Higgs bundles on . This generalises a result of Faltings from smooth projective curves to any higher dimension, and further to the rigid analytic setup. The strategy is new, and is based on the study of rigid analytic moduli spaces of pro-étale invertible sheaves on spectral varieties.
Paper Structure (24 sections, 42 theorems, 147 equations)

This paper contains 24 sections, 42 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. $p$-adic non-abelian Hodge theory
  4. Comparison to previous results
  5. Strategy
  6. The relative pro-étale Picard variety of the spectral cover
  7. Setup
  8. The relative pro-étale Picard variety of a coherent algebra
  9. The Leray exact sequence of the sheaf $\mathcal{B}^\times$
  10. The Higgs--Tate torsor of Abbes--Gros
  11. The partial splitting
  12. Right-exactness via the exponential map
  13. Representability of $\mathbf{Pic}_{\mathcal{B},{\operatorname{pro\acute{e}t}}}$
  14. Invertible $\mathcal{B}$-modules via the exponential
  15. Reduction of structure groups
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth proper rigid space over $K$. Then choices of a $\mathrm{B}_{\mathrm{dR}}^+/\xi^2$-lift $\mathbb X$ of $X$ and of an exponential $\mathrm{Exp}$ for $K$ induce an exact tensor equivalence which is natural in the datum of the pair $(X,\mathbb X)$.

Theorems & Definitions (109)

  • Theorem 1.1: \ref{['t:p-adicSimpson-proper']}
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: HMZ
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • ...and 99 more