Quasi-$F$-splitting versus log canonicity

In this paper, we investigate the relationship between quasi-$F$-splitting and log canonicity.

We show that if a numerically $\mathbb{Q}$-Gorenstein normal singularity is quasi-$F^e$-split for every $e\geq 1$, then it is numerically log canonical.

In dimension two, we prove the converse under the condition that the Gorenstein index is not divisible by the characteristic $p$.

We also classify two-dimensional quasi-$F$-split normal singularities.