On the abundance theorem for numerically trivial canonical divisors in positive characteristic

TL;DR

This paper proves the abundance theorem for numerically trivial canonical divisors on strongly F-regular varieties in positive characteristic, under the assumption of strong F-regularity of the geometric generic fibers of the Albanese morphism.

Contribution

It establishes the abundance theorem in positive characteristic for a new class of varieties with specific fiber regularity conditions.

Findings

Proves the abundance theorem for strongly F-regular varieties with certain fiber conditions.

Shows the importance of geometric generic fibers' regularity in the theorem.

Extends known results in the minimal model program to positive characteristic settings.

Abstract

In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly $F$-regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly $F$-regular.