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

Abstract

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.

arXiv comment31pages; v2: minor changes, Section 5 of v1 removed and incorporated into another paper

References in corpus (8)

• Globally $F$-regular and log Fano varieties
• Inversion of adjunction on log canonicity
• Singularities on normal varieties
• F-singularities of pairs and Inversion of Adjunction of arbitrary codimension
• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras
• An analog of adjoint ideals and PLT singularities in mixed characteristic
• Stability and deformation of F-singularities
• Linearly Reductive Quotient Singularities

Cited by in corpus (1)

• Torsors over the Rational Double Points in Characteristic $\mathbf{p}$