$F$-split and $F$-regular varieties with a diagonalizable group action

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.