Quasi-F-splittings in birational geometry

TL;DR

This paper develops the theory of quasi-$F$-splittings in birational geometry, providing criteria for liftability and applying it to three-dimensional klt singularities in large characteristic.

Contribution

It introduces the concept of quasi-$F$-splittings, establishes criteria using the higher Cartier operator, and proves that certain singularities are quasi-$F$-split.

Findings

Three-dimensional klt singularities in large characteristic are quasi-$F$-split.

Quasi-$F$-splittings imply liftability modulo p^2.

Criteria for quasi-$F$-splitting using the higher Cartier operator.

Abstract

We develop the theory of quasi-$F$-splittings in the context of birational geometry. Amongst other things, we obtain results on liftability of sections and establish a criterion for whether a scheme is quasi-$F$-split employing the higher Cartier operator. As one of the applications of our theory, we prove that three-dimensional klt singularities in large characteristic are quasi-$F$-split, and so, in particular, they lift modulo $p^2$.