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NOTE ON THE THREE-DIMENSIONAL LOG CANONICAL ABUNDANCE IN CHARACTERISTIC $>3$

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  28 February 2024

ZHENG XU Open the ORCID record for ZHENG XU [Opens in a new window]
ZHENG XU*
Affiliation:
Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China
*
[email protected]
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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.

Keywords

abundance conjecturethreefoldspositive characteristic

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14J30: $3$-folds 14E05: Rational and birational maps
Type
Article
Information
Nagoya Mathematical Journal , Volume 255 , September 2024 , pp. 694 - 723
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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