On the characterization of abelian varieties for log pairs in zero and positive characteristic
Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2515-2539.

Let (X,Δ) be a pair. We study how the values of the log Kodaira dimension and log plurigenera relates to surjectivity and birationality of the Albanese map and the Albanese morphism of X in both characteristic 0 and characteristic p>0. In particular, we generalize some well known results for smooth varieties in both zero and positive characteristic to varieties with various types of singularities. Moreover, we show that if X is a normal projective threefold in characteristic p>0, the coefficients of the components of Δ are 1 and -(K X +Δ) is semiample, then the Albanese morphism of X is surjective under certain assumptions on p and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analogue in dimension 3 of a result of Zhang on a conjecture of Demailly–Peternell–Schneider.

Soit (X,Δ) une paire. Nous étudions comment la dimension de Kodaira logarithmique et les plurigenres logarithmiques contrôlent la surjectivité et la birationalité de l’application d’Albanese ou du morphisme d’Albanese en caractéristique positive ou nulle. En particulier, nous généralisons certains résultats bien connus pour les variétés lisses à la fois en caractéristique zéro et caractéristique positive sous certaines variétés avec différents types de singularités. En outre, nous montrons que si X est une variété projective normale de dimension 3 en caractéristique p>0, les coefficients des composantes de Δ sont 1 et -(K X +Δ) est semiample, alors le morphisme Albanese de X est surjectif sous des certaines hypothèses sur p et les singularités des fibres générales du morphisme Albanese. C’est un analogue en caractéristique positive et en dimension 3 d’un résultat de Zhang sur une conjecture de Demailly–Peternell–Schneider.

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DOI: 10.5802/aif.3525
Classification: 14E15,  14J10,  14J17,  14J30
Keywords: abelian variety, Albanese map, Albanese morphism, canonical bundle formula, generic vanishing, Kodaira dimension, positive characteristic
Wang, Yuan 1

1 Northwestern University Department of Mathematics 2033 Sheridan Road Evanston, 60208 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Wang, Yuan. On the characterization of abelian varieties for log pairs in zero and positive characteristic. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2515-2539. doi : 10.5802/aif.3525. https://aif.centre-mersenne.org/articles/10.5802/aif.3525/

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