Chow ring and gonality of general abelian varieties
Annales Henri Lebesgue, Volume 1 (2018), pp. 313-332.

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree k has dimension at most k-1. Building on the work of Pirola, we show that very general abelian varieties of dimension g have (covering) gonality at least f(g), where f(g) grows like log g. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties A of dimension g, e.g., if g2k-1, the set of divisors D Pic 0 (A) such that D k =0 in CH k (A) is at most countable.

Nous étudions la gonalité des variétés abéliennes ainsi que leurs orbites de zéro-cycles pour l’équivalence rationnelle. Nous montrons que l’orbite d’un zéro-cycle de degré k est de dimension au plus k-1. En développant des idées de Pirola, nous montrons qu’une variété abélienne très générale a une gonalité au moins égale à f(g), où f(g) croît comme log g. Ceci répond à une question posée par Bastianelli, De Poi, Ein, Lazarsfeld et B. Ullery. Nous obtenons aussi des résultats sur l’anneau de Chow des variétés abéliennes A de dimension g ; par exemple, si g2k-1, l’ensemble des diviseurs D Pic 0 (A) tels que D k =0 dans CH k (A) est au plus dénombrable.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/ahl.10
Classification: 14K12, 14C25
Keywords: Abelian varieties, covering gonality, zero-cycles, Chow ring
Voisin, Claire 1

1 Collège de France 3 rue d’Ulm 75005 Paris (France)
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Voisin, Claire. Chow ring and gonality of general abelian varieties. Annales Henri Lebesgue, Volume 1 (2018), pp. 313-332. doi : 10.5802/ahl.10. http://www.numdam.org/articles/10.5802/ahl.10/

[AP93] Alzati, Alberto; Pirola, Gian P. Rational orbits on three-symmetric products of abelian varieties, Trans. Am. Math. Soc., Volume 337 (1993) no. 2, pp. 965-980 | DOI | MR | Zbl

[BDPE + 17] Bastianelli, Francesco; De Poi, Pietro; Ein, Lawrence; Lazarsfeld, Robert; Ullery, Brooke Measures of irrationality for hypersurfaces of large degree, Compos. Math., Volume 153 (2017) no. 11, pp. 2368-2393 | DOI | MR | Zbl

[Bea82] Beauville, Arnaud Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, Algebraic geometry (Tokyo/Kyoto, 1982) (Lecture Notes in Mathematics), Volume 1016, Springer, 1982, pp. 238-260 | DOI | Zbl

[Bea83] Beauville, Arnaud Variétés kählériennes dont la première classe de Chern est nulle, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 755-782 | DOI | Zbl

[Blo76] Bloch, Spencer Some elementary theorems about algebraic cycles on Abelian varieties, Invent. Math., Volume 37 (1976) no. 3, pp. 215-228 | DOI | MR | Zbl

[CvG93] Colombo, Elisabetta; van Geemen, Bert Note on curves in a Jacobian, Compos. Math., Volume 88 (1993) no. 3, pp. 333-353 | Numdam | MR | Zbl

[Her07] Herbaut, Fabien Algebraic cycles on the Jacobian of a curve with a linear system of given dimension, Compos. Math., Volume 143 (2007) no. 4, pp. 883-899 | DOI | MR | Zbl

[Huy14] Huybrechts, Daniel Curves and cycles on K3 surfaces, Algebr. Geom., Volume 1 (2014) no. 1, pp. 69-106 | MR | Zbl

[Mat58] Mattuck, Arthur Cycles on abelian varieties, Proc. Am. Math. Soc., Volume 9 (1958), pp. 88-98 | DOI | MR | Zbl

[Mum69] Mumford, David Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ., Volume 9 (1969), pp. 195-204 | DOI | MR | Zbl

[MZ17] Marian, Alina; Zhao, Xiaolei On the group of zero-cycles of holomorphic symplectic varieties (2017) (https://arxiv.org/abs/1711.10045)

[Pir89] Pirola, Gian P. Curves on generic Kummer varieties, Duke Math. J., Volume 59 (1989), pp. 701-708 | DOI | MR | Zbl

[Pir95] Pirola, Gian P. Abel-Jacobi invariant and curves on generic abelian varieties, Abelian varieties (Egloffstein, 1993), Walter de Gruyter, 1995, pp. 237-249 | MR | Zbl

[Voi92] Voisin, Claire Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 19 (1992) no. 4, pp. 473-492 | Zbl

[Voi15a] Voisin, Claire Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady, Recent advances in algebraic geometry (London Mathematical Society Lecture Note Series), Volume 417, Cambridge University Press, 2015, pp. 422-436 | DOI | MR | Zbl

[Voi15b] Voisin, Claire Some new results on modified diagonals, Geom. Topol., Volume 19 (2015) no. 6, pp. 3307-3343 | DOI | MR | Zbl

[Voi16] Voisin, Claire Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady, K3 surfaces and their moduli (Progress in Mathematics), Birkhäuser/Springer, 2016, pp. 365-399 | Zbl

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