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We prove a variant of the Beauville--Bogomolov decomposition for weaklyordinary, or generally globally $F$-split, varieties $X$ with $K_X \sim 0$, incharacteristic $p>0$. We also show that the weakly ordinary assumption in ourstatement cannot be dropped. Additionally, if the assumption $K_X \sim 0$ isreplaced by $-K_X$ being semi-ample, we show the weaker statement that allclosed fibers of the Albanese morphism are isomorphic. In the appendix writtenjointly with Giulio Codogni, we also deduce the singular version of the latterstatement, in characteristic zero. Finally, we apply our main theorem to drawconsequences to the behavior of rational points and fundamental groups ofweakly ordinary $K$-trivial varieties in positive characteristic.

On the Beauville--Bogomolov decomposition in characteristic $p\geq 0$