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

TL;DR

This paper extends the Beauville--Bogomolov decomposition to certain varieties in positive characteristic, revealing new structural insights and implications for rational points and fundamental groups.

Contribution

It proves a variant of the Beauville--Bogomolov decomposition for weakly ordinary varieties with trivial canonical class in characteristic p>0, and explores related geometric properties.

Findings

Weakly ordinary varieties with K_X ~ 0 satisfy a decomposition similar to characteristic zero.

The weakly ordinary assumption is essential; it cannot be omitted.

Applications include understanding rational points and fundamental groups of K-trivial varieties in positive characteristic.

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.