NOTE ON THE THREE-DIMENSIONAL LOG CANONICAL ABUNDANCE IN CHARACTERISTIC

Abstract

In this paper, we prove the nonvanishing 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 nontrivial. Moreover, we prove that the abundance for klt threefold pairs over k implies the abundance for log canonical threefold pairs over k.