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p-Curvature and Non-Abelian Cohomology

Yeuk Hay Joshua Lam, Daniel Litt

TL;DR

The paper develops a non-abelian analogue of Katz’s p-curvature theory for the Gauss–Manin connection by replacing cohomology with non-abelian cohomology and moduli of flat bundles. It introduces non-abelian filtrations (Hodge, conjugate) on the de Rham and Dolbeault moduli stacks, defines a non-abelian Gauss–Manin connection (isomonodromy), and proves a non-abelian Katz formula linking the p-curvature of the conjugate foliation to a Frobenius pullback of a Kodaira–Spencer-type lifting. Vanishing p-curvature for infinitely many primes implies vanishing of the non-abelian Higgs–Kodaira–Spencer data in characteristic zero, which yields unitary, relatively compact, and tri-holomorphic monodromy actions on Betti character varieties; this underpins finiteness results for non-abelian monodromy and several instances of the Ekedahl–Shepherd-Barron–Taylor conjecture, including new cases when $r=2$, the Betti moduli are irreducible, or $X/S$ is a relative curve. The work blends non-abelian Hodge theory in positive characteristic, deformation theory via $\\lambda$-Atiyah complexes, and complex-analytic hyperkähler geometry to control the arithmetic of non-abelian local systems. $

Abstract

Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on $H^i(X_s, \mathbb{Z})$ factors through a finite group. We prove a non-abelian analogue of this statement: if the $p$-curvature of the isomonodromy foliation on the moduli of flat bundles of rank $r$ on $X/S$ vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on the rank $r$ integral characters of $π_1(X_s)$ factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture. The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem.

p-Curvature and Non-Abelian Cohomology

TL;DR

The paper develops a non-abelian analogue of Katz’s p-curvature theory for the Gauss–Manin connection by replacing cohomology with non-abelian cohomology and moduli of flat bundles. It introduces non-abelian filtrations (Hodge, conjugate) on the de Rham and Dolbeault moduli stacks, defines a non-abelian Gauss–Manin connection (isomonodromy), and proves a non-abelian Katz formula linking the p-curvature of the conjugate foliation to a Frobenius pullback of a Kodaira–Spencer-type lifting. Vanishing p-curvature for infinitely many primes implies vanishing of the non-abelian Higgs–Kodaira–Spencer data in characteristic zero, which yields unitary, relatively compact, and tri-holomorphic monodromy actions on Betti character varieties; this underpins finiteness results for non-abelian monodromy and several instances of the Ekedahl–Shepherd-Barron–Taylor conjecture, including new cases when , the Betti moduli are irreducible, or is a relative curve. The work blends non-abelian Hodge theory in positive characteristic, deformation theory via -Atiyah complexes, and complex-analytic hyperkähler geometry to control the arithmetic of non-abelian local systems. $

Abstract

Let be a smooth projective morphism. Katz proved the Grothendieck-Katz -curvature conjecture for the Gauss-Manin connection on the -th cohomology of : if its -curvature vanishes mod for infinitely many , then the action of on factors through a finite group. We prove a non-abelian analogue of this statement: if the -curvature of the isomonodromy foliation on the moduli of flat bundles of rank on vanishes mod for infinitely many , then the action of on the rank integral characters of factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture. The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem.
Paper Structure (49 sections, 46 theorems, 153 equations, 1 table)

This paper contains 49 sections, 46 theorems, 153 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results (applications)
  3. Main structural results on non-abelian cohomology
  4. Katz's proof of the $p$-curvature conjecture for Gauss-Manin connections---the abelian setting
  5. A non-abelian analogue---Leitfaden
  6. Questions
  7. Related works
  8. Outline
  9. Acknowledgments
  10. The non-abelian Hodge and conjugate filtrations
  11. Flat bundles and the de Rham moduli stack
  12. The Dolbeault moduli stack
  13. The Hitchin morphism
  14. Filtrations on non-abelian cohomology
  15. The Hodge moduli stack
  16. ...and 34 more sections

Key Result

Theorem 1.1.1

Let $R\subset \mathbb{C}$ be a finitely-generated $\mathbb{Z}$-algebra, and let $f: X\to S$ be a smooth projective morphism of smooth $R$-schemes. Fix an integer $i\geq 0$ and denote by $(\mathscr{E}, \nabla):= (R^if_*\Omega_{X/S, \operatorname{dR}}^\bullet, \nabla_{GM})$ the $i$-th relative de Rham

Theorems & Definitions (131)

  • Theorem 1.1.1: Katz katz-p-curvature
  • Theorem A
  • Conjecture 1.1.2: Grothendieck-Katz
  • Conjecture 1.1.3: Ekedahl--Shepherd-Barron--Taylor esbt, Bost bost-foliations
  • Theorem B
  • Remark 1.1.4
  • Theorem C: Non-abelian Katz formula
  • Remark 1.2.3
  • Theorem D: Non-abelian Hodge index theorem I
  • Theorem E: Non-abelian Hodge index theorem II
  • ...and 121 more