On the canonical bundle formula and log abundance in positive characteristic

TL;DR

This paper proves a weak canonical bundle formula for fibrations of relative dimension one and applies it to establish key conjectures like log non-vanishing and log abundance for three-dimensional varieties over fields of positive characteristic.

Contribution

It introduces a weak canonical bundle formula in positive characteristic and uses it to prove the log non-vanishing and log abundance conjectures for threefolds.

Findings

Proves a weak canonical bundle formula for fibrations of relative dimension one.

Establishes the log non-vanishing conjecture for three-dimensional klt pairs in characteristic p>5.

Proves the log abundance conjecture for threefolds with non-maximal nef dimension.

Abstract

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field $k$ of characteristic $p>5$. We also show the log abundance conjecture for threefolds over $k$ when the nef dimension is not maximal, and the base point free theorem for threefolds over the algebraic closure of any finite field of characteristic $p>2$.