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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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 
@article{AIF_2022__72_6_2515_0,
     author = {Wang, Yuan},
     title = {On the characterization of abelian varieties for log pairs in zero and positive characteristic},
     journal = {Annales de l'Institut Fourier},
     pages = {2515--2539},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {6},
     year = {2022},
     doi = {10.5802/aif.3525},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3525/}
}
TY  - JOUR
AU  - Wang, Yuan
TI  - On the characterization of abelian varieties for log pairs in zero and positive characteristic
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 2515
EP  - 2539
VL  - 72
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3525/
DO  - 10.5802/aif.3525
LA  - en
ID  - AIF_2022__72_6_2515_0
ER  - 
%0 Journal Article
%A Wang, Yuan
%T On the characterization of abelian varieties for log pairs in zero and positive characteristic
%J Annales de l'Institut Fourier
%D 2022
%P 2515-2539
%V 72
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3525/
%R 10.5802/aif.3525
%G en
%F AIF_2022__72_6_2515_0
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/

[1] Blickle, Manuel; Schwede, Karl p -1 -linear maps in algebra and geometry, Commutative algebra, Springer, 2013, pp. 123-205 | Zbl

[2] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-630

[3] Chen, Jungkai A.; Hacon, Christopher D. Characterization of abelian varieties, Invent. Math., Volume 143 (2001) no. 2, pp. 435-447

[4] Chen, Jungkai A.; Hacon, Christopher D. On algebraic fiber spaces over varieties of maximal Albanese dimension, Duke Math. J., Volume 111 (2002) no. 1, pp. 159-175

[5] Chen, Jungkai A.; Hacon, Christopher D. On Ueno’s conjecture K, Math. Ann., Volume 345 (2009) no. 2, pp. 287-296

[6] Chen, Yifei; Zhang, Lei The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, Math. Res. Lett., Volume 22 (2015) no. 3, pp. 675-696

[7] Das, Omprokash; Schwede, Karl The F-different and a canonical bundle formula, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 3, pp. 1173-1205

[8] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Kähler manifold with numerically effective Ricci class, Compos. Math., Volume 89 (1993) no. 2, pp. 217-240

[9] Ein, Lawrence; Lazarsfeld, Robert Singularities of theta divisors and the birational geometry of irregular varieties, J. Am. Math. Soc., Volume 10 (1997) no. 1, pp. 243-258

[10] Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, 123, American Mathematical Society, 2005

[11] Hacon, Christopher D. A derived category approach to generic vanishing, J. Reine Angew. Math., Volume 575 (2004), pp. 173-187

[12] Hacon, Christopher D.; Kovács, Sándor Classification of higher dimensional algebraic varieties, Oberwolfach Seminars, 41, Birkhäuser, 2010

[13] Hacon, Christopher D.; Patakfalvi, Zsolt Generic vanishing in characteristic p>0 and the characterization of ordinary Abelian varieties, Am. J. Math., Volume 138 (2016) no. 4, pp. 963-998

[14] Hacon, Christopher D.; Patakfalvi, Zsolt On the characterization of abelian varieties in characteristic p>0 (2017) (https://arxiv.org/abs/1602.01791)

[15] Hacon, Christopher D.; Patakfalvi, Zsolt; Zhang, Lei Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0 (2017) (https://arxiv.org/abs/1703.06631)

[16] Hacon, Christopher D.; Popa, Mihnea; Schnell, Christian Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 143-195

[17] Hacon, Christopher D.; Xu, Chenyang On the three dimensional minimal model program in positive characteristics, J. Am. Math. Soc., Volume 28 (2015) no. 3, pp. 711-744

[18] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977

[19] Kawamata, Yujiro Characterization of abelian varieties, Compos. Math., Volume 43 (1981) no. 2, pp. 253-276

[20] Kawamata, Yujiro Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math., Volume 363 (1985), pp. 1-46

[21] Kollár, János Shafarevich Maps and Automorphic Forms, M. B. Porter Lectures, Princeton University Press, 1995

[22] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013

[23] Lang, Serge Abelian varieties, Springer, 1983

[24] Mochizuki, Shinichi Topics in absolute anabelian geometry I: generalities, J. Math. Sci., Tokyo, Volume 19 (2012) no. 2, pp. 139-242

[25] Moonen, Ben; van der Geer, Gerard Abelian Variety (preliminary version, available on the authors’ websites https://www.math.ru.nl/~bmoonen/research.html#bookabvar)

[26] Mukai, Shigeru Duality between D(X) and D(X ^) with its application to Picard sheaves, Nagoya Math. J., Volume 81 (1981), pp. 135-175

[27] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004

[28] Pareschi, Giuseppe; Popa, Mihnea GV-sheaves, Fourier–Mukai transform, and generic vanishing, Am. J. Math., Volume 133 (2011) no. 1, pp. 235-271

[29] Patakfalvi, Zsolt Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 5, pp. 991-1025

[30] Patakfalvi, Zsolt On subadditivity of Kodaira dimension in positive characteristic, J. Algebr. Geom., Volume 27 (2018) no. 1, pp. 21-53

[31] Patakfalvi, Zsolt; Schwede, Karl; Tucker, Kevin Notes for the workshop on positive characteristic algebraic geometry (2014) (https://arxiv.org/abs/1412.2203v1)

[32] Patakfalvi, Zsolt; Schwede, Karl; Zhang, Wenliang F-singularities in families (2013) (https://arxiv.org/abs/1305.1646v1)

[33] Popa, Mihnea; Schnell, Christian On direct images of pluricanonical bundles, Algebra Number Theory, Volume 8 (2014) no. 9, pp. 2273-2295

[34] Reid, Miles Projective morphisms according to Kawamata (1983) (unpublished manuscript)

[35] Schwede, Karl F-adjunction, Algebra Number Theory, Volume 3 (2009) no. 8, pp. 907-950

[36] Schwede, Karl; Tucker, Kevin A survey of test ideals (2011) (https://arxiv.org/abs/1104.2000v2)

[37] Shibata, Takahiro On generic vanishing for pluricanonical bundles (2016) (https://arxiv.org/abs/1507.03294v4)

[38] Tanaka, Hiromu Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field (2015) (https://arxiv.org/abs/1503.01264v3)

[39] Wang, Yuan On relative rational chain connectedness of threefolds with anti-big canonical divisors in positive characteristics (2015) (https://arxiv.org/abs/1512.06189)

[40] Wang, Yuan Rational curves on hypersurfaces, Mich. Math. J., Volume 66 (2017) no. 3, pp. 625-635

[41] Zhang, Qi On projective manifolds with nef anticanonical bundles, J. Reine Angew. Math., Volume 478 (1996), pp. 57-60

[42] Zhang, Qi On projective varieties with nef anticanonical divisors, Math. Ann., Volume 332 (2005) no. 3, pp. 697-703

Cité par Sources :