F-Split and F-Regular Varieties with a Diagonalizable Group Action

TL;DR

This paper establishes a connection between F-splitting and F-regularity of varieties with diagonalizable group actions and their quotients, providing a new framework to analyze these properties in algebraic geometry.

Contribution

It introduces a method to determine F-splitting and F-regularity of varieties via associated quotient log pairs, extending understanding of group actions in positive characteristic.

Findings

F-splitting and F-regularity are equivalent for the variety and its quotient log pair.

The framework applies to complexity-one T-varieties and toric vector bundles.

Compatible splittings relate subvarieties of the original variety to those of the quotient.

Abstract

Let $H$ be a diagonalizable group over an algebraically closed field $k$ of positive characteristic, and $X$ a normal $k$-variety with an $H$-action. Under a mild hypothesis, e.g. $H$ a torus or $X$ quasiprojective, we construct a certain quotient log pair $(Y,\Delta)$ and show that $X$ is F-split (F-regular) if and only if the pair $(Y,\Delta)$ if F-split (F-regular). We relate splittings of $X$ compatible with $H$-invariant subvarieties to compatible splittings of $(Y,\Delta)$, as well as discussing diagonal splittings of $X$. We apply this machinery to analyze the F-splitting and F-regularity of complexity-one $T$-varieties and toric vector bundles, among other examples.