no. 6
On the characterization of abelian varieties for log pairs in zero and positive characteristic
[Sur la caractérisation des variétés abelian pour les paires logarithmiques en caractéristique zéro et positive]
Wang, Yuan1
1 Northwestern University Department of Mathematics 2033 Sheridan Road Evanston, 60208 (USA)
Annales de l'Institut Fourier, Tome 72 (2022) no. 6, pp. 2515-2539.

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.

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.

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Révisé le :
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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
Mot clés : variété abelian, cartographie Albanese, morphisme Albanese, formule de fibré canonique, annulation générique, dimension Kodaira, caractéristique positive
Wang, Yuan 1

1 Northwestern University Department of Mathematics 2033 Sheridan Road Evanston, 60208 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 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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