On the global and local geometry of quasi-$F$-split varieties with trivial canonical bundle

We solve certain questions related to the geometry and singularities of quasi-$F$-split varieties with trivial canonical bundle.

First, we prove that regular quasi-$F^{\infty}$-split varieties are not geometrically uniruled (this generalizes and significantly simplifies the earlier results of Patakfalvi and Zdanowicz) and have geometrically canonical singularities.

Second, we show that there exist quasi-$F$-split surfaces with trivial canonical bundle which are not quasi-$F^{\infty}$-split, answering negatively a question raised by Kawakami, Takamatsu, Tanaka, Witaszek, Yobuko and Yoshikawa.

Third, we show that normal quasi-$F$-split varieties with trivial canonical bundle are geometrically normal (this extends a result of Kawakami, Takamatsu and Yoshikawa), and finally we prove that quasi-$F^e$-pure normal varieties $X$ such that $mp^eK_X$ is Cartier for $m$ coprime to $p$ are log canonical, under a resolution of singularities hypothesis.