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 $p$ has normal $F$-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 two-dimensional strongly $F$-regular singularities are stable under equal characteristic deformations. Our results give an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto on deformations of linearly reductive quotient singularities.
Categories: math.AG math.AC