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On the Chow ring of some special Calabi–Yau varieties

Part of: Cycles and subschemes

Published online by Cambridge University Press:  11 May 2021

Robert Laterveer
Robert Laterveer*
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS – Université de Strasbourg, 7 Rue René Descartes, 67084Strasbourg CEDEX, France
*
e-mail: robert.laterveer@math.unistra.fr
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Abstract

We consider Calabi–Yau n-folds X arising from certain hyperplane arrangements. Using Fu–Vial’s theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of X generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin’s conjecture for X, and Voevodsky’s smash-nilpotence conjecture for odd-dimensional X.

Keywords

Algebraic cyclesChow groupmotiveBloch–Beilinson filtrationdistinguished cyclessection property

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 2 , June 2022 , pp. 308 - 327
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

Supported by ANR grant ANR-20-CE40-0023.

References

André, Y., Motifs de dimension finie d’après. In: Kimura, S.-I. and O’Sullivan, P. (eds.), Séminaire Bourbaki, Astérisque 299 Exp. No. 929, Vol. viii, 2003/2004, pp. 115–145.Google Scholar
André, Y., Une introduction aux motifs (motifs purs, motifs mixtes, périodes). In: Panoramas et Synthèses [Panoramas and syntheses], Vol. 17, Société Mathématique de France, Paris, 2004, xii+261 pp. ISBN: 2-85629-164-3Google Scholar
Beauville, A., Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273(1986), 647–651.CrossRefGoogle Scholar
Beauville, A. and Voisin, C., On the Chow ring of a K3 surface. J. Alg. Geom. 13(2004), 417–426.CrossRefGoogle Scholar
Bini, G., Laterveer, R., and Pacienza, G., Voisin’s conjecture for zero-cycles on Calabi–Yau varieties and their mirrors. Adv. Geom. 20(2020), no. 1, 91–108.CrossRefGoogle Scholar
Burek, D., Higher-dimensional Calabi–Yau manifolds of Kummer type. Math. Nachrichten. 293(2020), no. 4, 638–650.CrossRefGoogle Scholar
Cynk, S. and Hulek, K., Construction and examples of higher-dimensional modular Calabi–Yau manifolds. Can. Math. Bull. 50(2007), no. 4, 486–503.CrossRefGoogle Scholar
Cynk, S. and Kocel–Cynk, B., Classification of double octic Calabi–Yau threefolds with ${h}^{1,2}\le 1$ defined by an arrangement of eight planes. Commun. Contemp. Math. Vol. 22, No. 01, 1850082 (2020), 38 pp. doi.org/10.1142/S0219199718500827 CrossRefGoogle Scholar
Cynk, S. and Meyer, C., Geometry and arithmetic of certain double octic Calabi–Yau threefolds. Can. Math. Bull. 48(2005), no. 2, 180–194.CrossRefGoogle Scholar
Cynk, S. and Meyer, C., Modularity of some non-rigid double octic Calabi–Yau threefolds. Rocky Mountain J. Math. 38(2008), no. 6, 1937–1958.Google Scholar
Cynk, S., Schütt, M., and van Straten, D., Hilbert modularity of some double octic Calabi–Yau threefolds. J. Number Theory 210(2020), 313–332.CrossRefGoogle Scholar
Cynk, S. and van Straten, D., Periods of rigid double octic Calabi–Yau threefolds. Ann. Pol. Math. 123(2019), no. Part 1, 243–258.Google Scholar
Deninger, C. and Murre, J., Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422(1991), 201–219.Google Scholar
Dolgachev, I., Weighted projective varieties. In: Carrell, J. B. (ed.), Group actions and vector fields, (Vancouver, B.C., 1981), Lecture Notes in Math., 956, Springer, Berlin, Germany, 1982, pp. 34–71.CrossRefGoogle Scholar
Fu, L., Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau complete intersections. Adv. Math. 244(2013), 894–924.CrossRefGoogle Scholar
Fu, L. and Vial, C., Distinguished cycles on varieties with motive of abelian type and the section property. J. Alg. Geom. 29(2020), 53–107.CrossRefGoogle Scholar
Fulton, W., Intersection theory. Springer–Verlag Ergebnisse der Mathematik, Berlin Heidelberg New York and Tokyo, 1984.CrossRefGoogle Scholar
Gerkmann, R., Sheng, M., van Straten, D., and Zuo, K., On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}}^3$ . Math. Ann. 355(2013), 187–214.CrossRefGoogle Scholar
Ingalls, C. and Logan, A., On the Cynk–Hulek criterion for crepant resolutions of double covers. Preprint, 2020. arXiv:2006.14981v2 Google Scholar
Kahn, B. and Sebastian, R., Smash-nilpotent cycles on abelian 3-folds. Math. Res. Lett. 16(2009), 1007–1010.CrossRefGoogle Scholar
Kahn, B. and Sujatha, R., Birational motives I: pure birational motives. Ann. K-Theory. 1(2016), no. 4, 379–440.CrossRefGoogle Scholar
Kimura, S.-I., Chow groups are finite dimensional, in some sense. Math. Ann. 331(2005), no. 1, 173–201.CrossRefGoogle Scholar
Laterveer, R., Some results on a conjecture of Voisin for surfaces of geometric genus one. Boll. Unione Mat. Italiana 9(2016), no. 4, 435–452.CrossRefGoogle Scholar
Laterveer, R., Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds. Rend. Circ. Mat. Palermo 65(2016), no. 2, 333–344.CrossRefGoogle Scholar
Laterveer, R., Algebraic cycles and Todorov surfaces. Kyoto J. Math. 58(2018), no. 3, 493–527.Google Scholar
Laterveer, R., Some Calabi–Yau fourfolds verifying Voisin’s conjecture. Ric. di Mat. 67(2018), no. 2, 401–411.CrossRefGoogle Scholar
Laterveer, R., Zero-cycles on self-products of varieties: some elementary examples verifying Voisin’s conjecture, Bolletino Unione Mat. Italiana 14 no. 2 (2021), 323–329, https://doi.org/10.1007/s40574-020-00259-0 CrossRefGoogle Scholar
Laterveer, R. and Vial, C., On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder varieties. Can. J. Math. 72(2020), no. 2, 505–536.CrossRefGoogle Scholar
Matsumoto, K., Sasaki, T., and Yoshida, M., The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type (3, 6). Int. J. Math. 3(1992), no. 1, 1–164.CrossRefGoogle Scholar
Matsumoto, K. and Terasoma, T., Arithmetic-geometric means for hyperelliptic curves and Calabi-Yau varieties. Int. J. Math. 21(2010), no. 7, 939–949.CrossRefGoogle Scholar
Meyer, C., Modular Calabi–Yau threefolds. Fields Institute Monographs, 22. American Mathematical Society, Providence, RI, 2005.Google Scholar
Murre, J., Nagel, J. and Peters, C., Lectures on the theory of pure motives. Amer. Math. Soc. University Lecture Series, 61, American Mathematical Society, Providence, RI, 2013.Google Scholar
O’Sullivan, P., Algebraic cycles on an Abelian variety. J. Reine Angew. Math. 654(2011), 1–81.CrossRefGoogle Scholar
Paranjape, K., Abelian varieties associated to certain K3 surfaces. Comp. Math. 68(1988), no. 1, 11–22.Google Scholar
Scholl, T., Classical motives. In: Jannsen, U. et al., (eds.), Motives, Proceedings of Symposia in Pure Mathematics, 55, 1994, Part 1.CrossRefGoogle Scholar
Shen, M. and Vial, Ch., The Fourier transform for certain hyperKähler fourfolds. Memoirs of the AMS 240 (2016), no.1139.Google Scholar
Terasoma, T., Complete intersections of hypersurfaces—the Fermat case and the quadric case. Japan J. Math. (N.S.) 14(1988), no. 2, 309–384.Google Scholar
Terasoma, T., Algebraic correspondences between genus three curves and certain Calabi–Yau varieties. Amer. J. Math. 132(2010), no. 1, 181–200.Google Scholar
Vial, C., Generic cycles, Lefschetz representations and the generalized Hodge and Bloch conjectures for abelian varieties. Annali della Scuola Normale Superiore di Pisa 21(2020), 1389–1429.Google Scholar
Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero. Int. Math. Res. Not. 4(1995), 187–198.CrossRefGoogle Scholar
Voisin, C., The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. Ann. Sci. Ecole Norm. Sup. 46(2013), no. 3, 449–475.CrossRefGoogle Scholar
Voisin, C., Chow rings, decomposition of the diagonal, and the topology of families. Princeton University Press, Princeton and Oxford, 2014.CrossRefGoogle Scholar
Voisin, C., Remarks on zero-cycles of self-products of varieties. In: Maruyama, M. (ed.), Moduli of vector bundles, Proceedings of the Taniguchi Congress Marcel Dekker, New York, Basel and Hong Kong, 1994.Google Scholar
Yoshida, M., Hypergeometric functions, my love: modular interpretations of configuration spaces. Aspects of Mathematics, E32. Friedr. Vieweg & Sohn, Braunschweig, 1997.CrossRefGoogle Scholar