On the characterization of abelian varieties for log pairs in zero and positive characteristic

DOI10.5802/AIF.3525arXiv1610.05630MaRDI QIDQ6278759

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Publication date: 18 October 2016

Abstract: Let $(X,Delta)$ be a pair. We study how the condition $kappa(K_X+Delta)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p>0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $Delta$ are $le1$ and $-(K_X+Delta)$ is semiample, then the Albanese morphism of $X$ is surjective under reasonable assumptions on $p$ and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analog in dimension 3 of a result of Zhang on a conjecture of Demailly-Peternell-Schneider.

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