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The weak Beauville--Bogomolov decomposition in characteristic $p\geq 0$

Zsolt Patakfalvi, Maciej Zdanowicz

TL;DR

The paper addresses how Beauville–Bogomolov-type decompositions, classically known in characteristic zero, translate to positive characteristic. It develops a weak form of the decomposition under weak ordinarity or global F-splitting for K_X ~ 0 varieties, showing the decomposition can occur after finite (often inseparable) covers and a quasi-étale or infinitesimal torsor, with V allowed to have strongly F-regular singularities. A central technical program combines Albanese-morphism analysis, positivity in families, test-ideals, and numerically flat bundles, plus finite-field reductions and Nori’s fundamental group machinery to produce isotriviality results and a robust structure theorem. The results extend to singular pair settings and yield consequences for rational points and fundamental groups in positive characteristic, while careful counterexamples demonstrate the necessity of the hypotheses. Collectively, the work significantly advances the understanding of the structure of K-trivial varieties in characteristic p and highlights new phenomena absent in characteristic zero, including infinitesimal torsors and finite-field-induced isotriviality.

Abstract

We prove a variant of the Beauville--Bogomolov decomposition for weakly ordinary, or generally globally $F$-split, varieties $X$ with $K_X \sim 0$, in characteristic $p>0$. We also show that the weakly ordinary assumption in our statement cannot be dropped. Additionally, if the assumption $K_X \sim 0$ is replaced by $-K_X$ being semi-ample, we show the weaker statement that all closed fibers of the Albanese morphism are isomorphic. Finally, we apply our main theorem to draw consequences to the behavior of rational points and fundamental groups of weakly ordinary $K$-trivial varieties in positive characteristic.

The weak Beauville--Bogomolov decomposition in characteristic $p\geq 0$

TL;DR

The paper addresses how Beauville–Bogomolov-type decompositions, classically known in characteristic zero, translate to positive characteristic. It develops a weak form of the decomposition under weak ordinarity or global F-splitting for K_X ~ 0 varieties, showing the decomposition can occur after finite (often inseparable) covers and a quasi-étale or infinitesimal torsor, with V allowed to have strongly F-regular singularities. A central technical program combines Albanese-morphism analysis, positivity in families, test-ideals, and numerically flat bundles, plus finite-field reductions and Nori’s fundamental group machinery to produce isotriviality results and a robust structure theorem. The results extend to singular pair settings and yield consequences for rational points and fundamental groups in positive characteristic, while careful counterexamples demonstrate the necessity of the hypotheses. Collectively, the work significantly advances the understanding of the structure of K-trivial varieties in characteristic p and highlights new phenomena absent in characteristic zero, including infinitesimal torsors and finite-field-induced isotriviality.

Abstract

We prove a variant of the Beauville--Bogomolov decomposition for weakly ordinary, or generally globally -split, varieties with , in characteristic . We also show that the weakly ordinary assumption in our statement cannot be dropped. Additionally, if the assumption is replaced by being semi-ample, we show the weaker statement that all closed fibers of the Albanese morphism are isomorphic. Finally, we apply our main theorem to draw consequences to the behavior of rational points and fundamental groups of weakly ordinary -trivial varieties in positive characteristic.

Paper Structure

This paper contains 54 sections, 75 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Historical motivation
  3. List of the results
  4. The smooth cases of the characteristic $p>0$ results
  5. Statements for singular varieties in characteristic $p>0$
  6. Outline of the proof
  7. Albanese morphism and augmented irregularity in characteristic $p$
  8. General fibers are strongly $F$-regular
  9. Flatness
  10. Restriction to curves
  11. Numerical flatness
  12. Isotriviality for finite fields
  13. Reduction to finite fields
  14. From isotriviality over curves to the isotriviality over $\mathop{\mathrm{Alb}}\nolimits_Z$ and to the splitting over a flat cover
  15. Finiteness of automorphism groups
  16. ...and 39 more sections

Key Result

Theorem 1.1

Let $X$ be smooth projective variety over $k$, such that $K_X \sim 0$ and $X$ is weakly ordinary. Then there is a composition of two finite covers, such that Additionally, the action of $G:=\prod_{i=1}^n \mu_{p^{j_i}}$ on $B \times V$ is the diagonal action induced by an action on $V$ and an action on $B$, respectively, such that

Theorems & Definitions (157)

  • Theorem 1.1: Smooth & weakly ordinary case of the Beauville--Bogomolov decomposition, special case of \ref{['thm:beauville_bogomolov_general_logcy']}
  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Theorem 1.2: Isotriviality over curves, special case of \ref{['cor:finite_field']} and \ref{['cor:semi_ample']}.
  • Theorem 1.3: Isotriviality of the Albanese morphism, special case of \ref{['thm:decomposition_theorem']}
  • Remark 4
  • Conjecture 1
  • Corollary 1: special case of \ref{['cor:rational_points']}
  • ...and 147 more