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On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

Published online by Cambridge University Press:  03 September 2019

Robert Laterveer
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS – Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX, France Email: robert.laterveer@math.unistra.fr
Charles Vial
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 10031, 33501 Bielefeld, Germany Email: vial@math.uni-bielefeld.de

Abstract

This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

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© Canadian Mathematical Society 2019

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References

André, Y., Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan, …). Séminaire Bourbaki 2003/2004, Astérisque 299(2005), viii, 115–145.Google Scholar
Bazhov, I., On the Chow group of zero–cycles of Calabi–Yau hypersurfaces. arxiv:1510.05516.Google Scholar
Beauville, A. and Voisin, C., On the Chow ring of a K3 surface. J. Algebraic Geom. 13(2004), 417426. https://doi.org/10.1090/S1056-3911-04-00341-8Google Scholar
Beauville, A., On the splitting of the Bloch–Beilinson filtration. In: Algebraic cycles and motives. London Math. Soc. Lecture Note Ser., 344, Cambridge University Press, Cambridge, 2007.Google Scholar
Beauville, A., Some surfaces with maximal Picard number. J. Éc. Polytech. Math. 1(2014), 101116. https://doi.org/10.5802/jep.5Google Scholar
Bini, G., Laterveer, R., and Pacienza, G., Voisin’s conjecture for zero-cycles on Calabi–Yau varieties and their mirrors. Adv. Geom., to appear.Google Scholar
Cynk, S. and Hulek, K., Construction and examples of higher-dimensional modular Calabi–Yau manifolds. Canad. Math. Bull. 50(2007), 486503. https://doi.org/10.4153/CMB-2007-049-9Google Scholar
Deninger, Ch. and Murre, J., Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422(1991), 201219. https://doi.org/10.1515/crll.1991.422.201Google Scholar
Fakhruddin, N., On the Chow groups of supersingular varieties. Canad. Math. Bull. 45(2002), 204212. https://doi.org/10.4153/CMB-2002-023-5Google Scholar
Flapan, L., Schreieder’s surfaces are elliptic modular. arxiv:1603.05613v1.Google Scholar
Flapan, L. and Lang, J., Chow motives associated to certain algebraic Hecke characters. Trans. Amer. Math. Soc. Ser. B 5(2018), 102124. https://doi.org/10.1090/btran/27Google Scholar
Fu, L., Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau complete intersections. Adv. Math. 244(2013), 894924. https://doi.org/10.1016/j.aim.2013.06.008Google Scholar
Fu, L., Tian, Z., and Vial, Ch., Motivic hyperKähler resolution conjecture: I. Generalized Kummer varieties. Geom. Topol. 23(2019), 427492. https://doi.org/10.2140/gt.2019.23.427Google Scholar
Fu, L. and Vial, Ch., Distinguished cycles on varieties with motive of abelian type and the section property. J. Algebraic Geom., to appear.Google Scholar
Fulton, W., Intersection theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1984. https://doi.org/10.1007/978-3-662-02421-8Google Scholar
Hulek, K., Kloosterman, R., and Schütt, M., Modularity of Calabi–Yau varieties. In: Global aspects of complex geometry. Springer, Berlin, 2006, pp. 271309.https://doi.org/10.1007/3-540-35480-8_8Google Scholar
Jannsen, U., Motivic sheaves and filtrations on Chow groups. In: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1. American Mathematical Society, Providence, RI, 1994.Google Scholar
Kahn, B., Equivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Ann. Sci. École Norm. Sup. 36(2003), no. 6, 9771002. https://doi.org/10.1016/j.ansens.2003.02.002Google Scholar
Kahn, B. and Sebastian, R., Smash-nilpotent cycles on abelian 3-folds. Math. Res. Lett. 16(2009), 10071010. https://doi.org/10.4310/MRL.2009.v16.n6.a8Google Scholar
Kimura, S., Chow groups are finite dimensional, in some sense. Math. Ann. 331(2005), 173201. https://doi.org/10.1007/s00208-004-0577-3Google Scholar
Kimura, S., Surjectivity of the cycle class map for Chow motives. Fields Inst. Commun., 56, American Mathematical Society, Providence, RI, 2009, pp. 157165.Google Scholar
Laterveer, R., Some results on a conjecture of Voisin for surfaces of geometric genus one. Boll. Unione Mat. Ital. 9(2016), 435452. https://doi.org/10.1007/s40574-016-0060-6Google Scholar
Laterveer, R., Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds. Rend. Circ. Mat. Palermo 65(2016), 333344. https://doi.org/10.1007/s12215-016-0237-yGoogle Scholar
Laterveer, R., Algebraic cycles and Todorov surfaces. Kyoto J. Math. 58(2018), 493527. https://doi.org/10.1215/21562261-2017-0027Google Scholar
Murre, J., On a conjectural filtration on the Chow groups of an algebraic variety. I and II. Indag. Math. 4(1993), 177201. https://doi.org/10.1016/0019-3577(93)90038-ZGoogle Scholar
O’Grady, K., Decomposable cycles and Noether–Lefschetz loci. Doc. Math. 21(2016), 661687.Google Scholar
O’Sullivan, P., Algebraic cycles on an abelian variety. J. Reine Angew. Math. 654(2011), 181. https://doi.org/10.1515/CRELLE.2011.025Google Scholar
Schreieder, S., On the construction problem for Hodge numbers. Geom. Topol. 19(2015), 295342. https://doi.org/10.2140/gt.2015.19.295Google Scholar
Shen, M. and Vial, Ch., The Fourier transform for certain hyperKähler fourfolds. Mem. Amer. Math. Soc. 240(2016), no. 1139. https://doi.org/10.1090/memo/1139Google Scholar
Shen, M. and Vial, Ch., On the motive of the Hilbert cube X [3]. Forum Math. Sigma 4(2016), e30. https://doi.org/10.1017/fms.2016.25Google Scholar
Shioda, T. and Inose, H., On singular K3 surfaces. In: Complex analysis and algebraic geometry. Iwanami Shoten, Tokyo, 1977, pp. 119136.Google Scholar
Vial, Ch., Pure motives with representable Chow groups. C. R. Math. Acad. Sci. Paris 348(2010), 11911195. https://doi.org/10.1016/j.crma.2010.10.017Google Scholar
Vial, Ch., On the motive of some hyperkähler varieties. J. für Reine u. Angew. Math. 725(2017), 235247. https://doi.org/10.1515/crelle-2015-0008Google Scholar
Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Not. 1995 no. 4, 187198. https://doi.org/10.1155/S1073792895000158Google Scholar
Voisin, C., Remarks on zero-cycles of self-products of varieties. In: Moduli of vector bundles (Sanda, 1994; Kyoto, 1994). Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996.Google Scholar
Voisin, C., Chow rings and decomposition theorems for K3 surfaces and Calabi–Yau hypersurfaces. Geom. Topol. 16(2012), 433473. https://doi.org/10.2140/gt.2012.16.433Google Scholar
Voisin, C., Chow rings, decomposition of the diagonal, and the topology of families. Princeton University Press, Princeton, NJ, 2014. https://doi.org/10.1515/9781400850532Google Scholar
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