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Logarithmic de Rham Stacks and Non-Abelian Hodge Theory

Michael Barz

TL;DR

The paper introduces logarithmic de Rham stacks X^{dR}(log D) and their sheared variants to geometrize logarithmic de Rham cohomology in characteristic p. It develops a robust relative de Rham framework, establishes a logarithmic Cartier descent via a stacky Frobenius twist, and constructs a logarithmic non-abelian Hodge theory for curves by relating Higgs bundles on a stacky Frobenius-twisted base to flat logarithmic connections, using Hitchin-type morphisms and spectral-data techniques. Central to the theory are the log Azumaya property of D-modules, p-curvature formalism, and a BeFNR-style correspondence augmented by residue/parabolic data; these yield an etale-local equivalence between de Rham and Dolbeault moduli and connect to de Cataldo–Zhang’s logarithmic NAH theorem. The work also develops root/multiroot stacks, residues, and a stack of splittings to organize the non-abelian correspondence, with extensive comparison to prior log-NAH results and an outlook on weight filtrations and future geometrization of logarithmic structures.

Abstract

In this article, we introduce the logarithmic de Rham stack of a pair (X, D), for a smooth variety X over a field k of positive characteristic p, and D a strict normal crossings divisor on X. Using this stack, we prove a new version of logarithmic Cartier descent, and a new logarithmic non-abelian Hodge theorem for curves, both stated using a certain logarithmic Frobenius twist. Our logarithmic non-abelian Hodge theorem implies an earlier logarithmic non-abelian Hodge theorem of de Cataldo-Zhang.

Logarithmic de Rham Stacks and Non-Abelian Hodge Theory

TL;DR

The paper introduces logarithmic de Rham stacks X^{dR}(log D) and their sheared variants to geometrize logarithmic de Rham cohomology in characteristic p. It develops a robust relative de Rham framework, establishes a logarithmic Cartier descent via a stacky Frobenius twist, and constructs a logarithmic non-abelian Hodge theory for curves by relating Higgs bundles on a stacky Frobenius-twisted base to flat logarithmic connections, using Hitchin-type morphisms and spectral-data techniques. Central to the theory are the log Azumaya property of D-modules, p-curvature formalism, and a BeFNR-style correspondence augmented by residue/parabolic data; these yield an etale-local equivalence between de Rham and Dolbeault moduli and connect to de Cataldo–Zhang’s logarithmic NAH theorem. The work also develops root/multiroot stacks, residues, and a stack of splittings to organize the non-abelian correspondence, with extensive comparison to prior log-NAH results and an outlook on weight filtrations and future geometrization of logarithmic structures.

Abstract

In this article, we introduce the logarithmic de Rham stack of a pair (X, D), for a smooth variety X over a field k of positive characteristic p, and D a strict normal crossings divisor on X. Using this stack, we prove a new version of logarithmic Cartier descent, and a new logarithmic non-abelian Hodge theorem for curves, both stated using a certain logarithmic Frobenius twist. Our logarithmic non-abelian Hodge theorem implies an earlier logarithmic non-abelian Hodge theorem of de Cataldo-Zhang.

Paper Structure

This paper contains 63 sections, 61 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Logarithmic Cartier descent
  3. Logarithmic non-abelian Hodge theory
  4. Relation to work of Li--Sun
  5. Relation to work of Ogus--Vologodsky and Schepler
  6. Weight filtrations
  7. Organization of this paper
  8. Acknowledgements
  9. The relative de Rham stack and its sheared variant
  10. The ring stacks $\mathbb{G}_a^{\mathop{\mathrm{dR}}\nolimits}$ and $\mathbb{G}_a^{\widehat{\mathop{\mathrm{dR}}\nolimits}}$
  11. Reminder on quasi-ideals
  12. The quasi-ideals $\mathbb{G}_a^{\#}$ and $\mathbb{G}_a^{\#, \wedge}$
  13. Witt vectors and $\mathbb{G}_a^{\#}$
  14. Sheared Witt vectors and $\mathbb{G}_a^{\#, \wedge}$
  15. Defining de Rham stacks
  16. ...and 48 more sections

Key Result

Theorem 1.1

The functor restricts to an equivalence of categories gives an equivalence of categories between $\mathop{\mathrm{Qcoh}}\nolimits(X')$ and the category of flat connections on $X$ with $p$-curvature 0.

Theorems & Definitions (200)

  • Definition 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.1: Cartier descent, Theorem 5.1 of katz
  • Example 1.1
  • Theorem 1.2: Logarithmic Cartier descent
  • Remark 1.5
  • ...and 190 more