Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities.

Given a normal $\mathbb{Q}$-Gorenstein complex variety $X$, we prove that if one spreads it out to a normal $\mathbb{Q}$-Gorenstein scheme $\mathcal{X}$ of mixed characteristic whose reduction $\mathcal{X}_p$ modulo $p$ is normal with $F$-pure singularities for a single prime $p$, then $X$ has log canonical singularities. In addition, we show its analog for log terminal singularities, without assuming that $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, which is a generalization of a result of Ma-Schwede. We also prove that strongly $F$-regular (resp. log canonical) singularities are stable under equal characteristic deformations if the nearby fibers are $\mathbb{Q}$-Gorenstein. Our results give an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto on deformations of linearly reductive quotient singularities.