The $F$-different and a canonical bundle formula

TL;DR

This paper explores the structure of Frobenius splittings on compatible subvarieties, relating them to the divisorial part of the different and providing a reinterpretation of the $F$-canonical bundle formula.

Contribution

It establishes bounds on divisors associated with Frobenius splittings using the divisorial part of the different and connects Frobenius splitting behavior with the $F$-canonical bundle formula.

Findings

Divisors from Frobenius splittings are bounded below by the divisorial part of the different.

The difference between the splitting divisor and the different is influenced by Frobenius splitting of fibers.

Reinterpretation of the $F$-canonical bundle formula through Frobenius splitting techniques.

Abstract

We study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties $W \subseteq X$. In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model $E \subseteq X' \to X$ (ie, this is a log canonical center), then we show that the divisor corresponding to the splitting on $W$ is bounded below by the divisorial part of the different as studied by Kawamata, Shokurov, Ambro and others. We also show that difference between the divisor associated to the splitting and the divisorial part of the different is largely governed by the (non-)Frobenius splitting of fibers of $E \to W$. In doing this analysis, we recover an $F$-canonical bundle formula by reinterpretting techniques common in the theory of Frobenius splittings.