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Given a normal $\mathbb{Q}$-Gorenstein complex variety $X$, we prove that ifone spreads it out to a normal $\mathbb{Q}$-Gorenstein scheme $\mathcal{X}$ ofmixed characteristic whose reduction $\mathcal{X}_p$ modulo $p$ has normal$F$-pure singularities for a single prime $p$, then $X$ has log canonicalsingularities. In addition, we show its analog for log terminal singularities,without assuming that $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, which is ageneralization of a result of Ma-Schwede. We also prove that two-dimensionalstrongly $F$-regular singularities are stable under equal characteristicdeformations. Our results give an affirmative answer to a conjecture ofLiedtke-Martin-Matsumoto on deformations of linearly reductive quotientsingularities.

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