Note on the 3-dimensional log canonical abundance in characteristic $>3$

TL;DR

This paper advances the understanding of the abundance conjecture for log canonical threefold pairs in characteristic greater than 3, establishing non-vanishing, semi-ampleness, and finite generation results.

Contribution

It proves non-vanishing and semi-ampleness of the canonical divisor for log canonical threefold pairs in characteristic p>3, and shows finite generation of their log canonical rings.

Findings

Non-vanishing of the canonical divisor when pseudo-effective.

Semi-ampleness when the divisor is nef and has positive Kodaira dimension.

Finite generation of log canonical rings in certain cases.

Abstract

In this paper, we prove the non-vanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field $k$ of characteristic $p > 3$. More precisely, we prove that if $(X,B)$ be a projective log canonical threefold pair over $k$ and $K_{X}+B$ is pseudo-effective, then $\kappa(K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $\kappa(K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample. As applications, we show that the log canonical rings of projective log canonical threefold pairs over $k$ are finitely generated and the abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm{Alb}(X)$ is non-trivial. Moreover, we prove that the abundance for klt threefold pairs over $k$ implies the abundance for log canonical threefold pairs over $k$.