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NOTE ON THE THREE-DIMENSIONAL LOG CANONICAL ABUNDANCE IN CHARACTERISTIC 
$>3$
Published online by Cambridge University Press: 28 February 2024
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.
MSC classification
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal
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