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ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

Part of: Cycles and subschemes

Published online by Cambridge University Press:  22 March 2021

CHIARA CAMERE ,
ALBERTO CATTANEO  and
ROBERT LATERVEER
CHIARA CAMERE
Affiliation:
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano (MI), Italy e-mail: chiara.camere@unimi.it
ALBERTO CATTANEO
Affiliation:
Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany e-mail: cattaneo@math.uni-bonn.de
ROBERT LATERVEER
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS – Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX, France e-mail: robert.laterveer@math.unistra.fr
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Abstract

We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 253 - 276
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Addington, N. and Lehn, M., On the symplectic eightfold associated to a Pfaffian cubic fourfold, J. Reine Angew. Math. 731 (2017), 129–137.Google Scholar
André, Y., Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan,…), Séminaire Bourbaki 2003/2004, Astérisque 299 (2005), Exp. No. 929, viii, 115–145.Google Scholar
Beauville, A. and Donagi, R., La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301(14) (1985), 703–706.Google Scholar
Beauville, A., On the splitting of the Bloch–Beilinson filtration, in Algebraic cycles and motives (Nagel, J. and Peters, C., Editors), London Math. Soc. Lecture Notes 344, Cambridge University Press, 2007.Google Scholar
Beauville, A. and Voisin, C., On the Chow ring of a K3 surface, J. Alg. Geom. 13 (2004), 417–426.CrossRefGoogle Scholar
Bloch, S., Lectures on algebraic cycles (Duke University Press, 1980).Google Scholar
Bloch, S. and Ogus, A., Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 4 (1974), 181–202.CrossRefGoogle Scholar
Bloch, S. and Srinivas, V., Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (5) (1983), 1235–1253.CrossRefGoogle Scholar
Boissière, S., Camere, C. and Sarti, A., Classification of automorphisms on a deformation family of hyper-Kähler four-folds by p-elementary lattices, Kyoto J. Math. 56(3) (2016), 465–499.CrossRefGoogle Scholar
Boissière, S., Camere, C. and Sarti, A., Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball, Math. Ann. 373(3–4) (2019), 1429–1455.CrossRefGoogle Scholar
Buckley, A. and Košir, T., Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115–140.CrossRefGoogle Scholar
Bülles, T.-H., Motives of moduli spaces on K3 surfaces and of special cubic fourfolds, Manuscripta Math. 161(1–2) (2020), 109–124.CrossRefGoogle Scholar
Camere, C. and Cattaneo, A., Non-symplectic automorphisms of odd prime order on manifolds of K3[n]-type, Manuscripta Math. 163 (2020), 299–342.CrossRefGoogle Scholar
de Cataldo, M. and Migliorini, L., The Chow groups and the motive of the Hilbert scheme of points on a surface, J. Algebra 251(2) (2002), 824–848.CrossRefGoogle Scholar
Charles, F. and Markman, E., The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces, Compos. Math. 149(3) (2013), 481–494.CrossRefGoogle Scholar
Chen, H., The Voisin map via families of extensions, arXiv:1806.05771 (2018).Google Scholar
Debarre, O., Hyperkähler manifolds, arXiv:1810.02087 (2018).Google Scholar
Dolgachev, I., Classical algebraic geometry. A modern view (Cambridge University Press, 2012).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
Floccari, S., Fu, L. and Zhang, Z., On the motive of O’Grady’s 10-dimensional hyper-Kähler varieties, Commun. Contemp. Math. (2020). https://doi.org/10.1142/S0219199720500340.Google Scholar
Fu, L., Laterveer, R. and Vial, Ch, The generalized Franchetta conjecture for some hyper–Kähler varieties (with an appendix joint with M. Shen), J. Math. Pures Appl. 130(9) (2019), 1–35.CrossRefGoogle Scholar
Fu, L., Tian, Z. and Vial, Ch, Motivic hyperkähler resolution conjecture, I: Generalized Kummer varieties, Geom. Topol., 23(1) (2019), 427–492.CrossRefGoogle Scholar
Fulton, W., Intersection theory (Springer–Verlag Ergebnisse der Mathematik, 1984).CrossRefGoogle Scholar
Hassett, B., Special cubic fourfolds, Compos. Math. 120(1) (2000), 1–23.CrossRefGoogle Scholar
Huybrechts, D., Curves and cycles on K3 surfaces, Algebr. Geom. 1(1) (2014), 69–106.CrossRefGoogle Scholar
Jannsen, U., Motivic sheaves and filtrations on Chow groups, in Motives (Jannsen, U. et al., eds.), Proc. Sympos. Pure Math., 55(Part 1) (1994), Amer. Math. Soc.Google Scholar
Jannsen, U., On finite-dimensional motives and Murre’s conjecture, in Algebraic cycles and motives (J. Nagel and C. Peters, Editors) (Cambridge University Press, 2007).Google Scholar
Kimura, S., Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173–201.CrossRefGoogle Scholar
Kollár, J., Laza, R., Saccà, G. and Voisin, C., Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier (Grenoble) 68(7) (2018), 2837–2882.CrossRefGoogle Scholar
Lahoz, M., Lehn, M., Macrì, E. and Stellari, P., Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects, J. Math. Pures Appl. 114 (2018), 85–117.CrossRefGoogle Scholar
Laterveer, R., A remark on the motive of the Fano variety of lines on a cubic, Ann. Math. Québec 41(1) (2017), 141–154.CrossRefGoogle Scholar
Laterveer, R., A family of cubic fourfolds with finite-dimensional motive, Journal Math. Soc. Japan 70(4) (2018), 1453–1473.CrossRefGoogle Scholar
Laterveer, R., Algebraic cycles on certain hyperkähler fourfolds with an order 3 non-symplectic automorphism, North–Western European J. Math. 4 (2018), 99–118.Google Scholar
Laterveer, R., Algebraic cycles and EPW cubes, Math. Nachr. 291(7) (2018), 1088–1113.CrossRefGoogle Scholar
Laterveer, R. and Vial, Ch, On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder varieties, Canadian J. Math. 72(2) (2020), 505–536.CrossRefGoogle Scholar
Lehn, C., Twisted cubics on singular cubic fourfolds – On Starr’s fibration, Math. Z. 290(1–2) (2018), 379–388.CrossRefGoogle Scholar
Lehn, M., Twisted cubics on a cubic fourfold and an involution on the associated 8-dimensional symplectic manifold, Oberwolfach Report No. 51/2015.Google Scholar
Lehn, C., Lehn, M., Sorger, C., and van Straten, D., Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87–128.Google Scholar
Li, C., Pertusi, L. and Zhao, X., Twisted cubics on cubic fourfolds and stability conditions, arXiv:1802.01134,Google Scholar
Muratore, G., The indeterminacy locus of the Voisin map, Beitr. Algebra Geom. 61(1) (2020), 73–88.CrossRefGoogle Scholar
Murre, J., On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag. Math. 4 (1993), 177–201.CrossRefGoogle Scholar
Murre, J., Nagel, J. and Peters, C., Lectures on the theory of pure motives, Amer. Math. Soc. University Lecture Series 61 (2013).CrossRefGoogle Scholar
Otwinowska, A., Remarques sur les groupes de Chow des hypersurfaces de petit degré, C. R. Acad. Sci. Paris Sr. I Math. 329(1) (1999), 51–56.CrossRefGoogle Scholar
Ouchi, G., Lagrangian embeddings of cubic fourfolds containing a plane, Compos. Math. 153(5) (2017), 947–972.CrossRefGoogle Scholar
Pan, X., Automorphism and cohomology I: Fano variety of lines and cubic, Alg. Geometry, doi: 10.14231/AG-2020-001.Google Scholar
Rieß, U., On the Chow ring of birational irreducible symplectic varieties, Manuscripta Math. 145 (2014), 473–501.CrossRefGoogle Scholar
Scholl, T., Classical motives, in Motives (Jannsen, U. et al., eds.), Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc. (1994).CrossRefGoogle Scholar
Shen, M. and Vial, Ch, The Fourier transform for certain hyperKähler fourfolds, Memoirs AMS 240(1139) (2016).CrossRefGoogle Scholar
Shen, M. and Vial, Ch, The motive of the Hilbert cube X[3] , Forum Math. Sigma 4 (2016), 55 pp.CrossRefGoogle Scholar
Shinder, E. and Soldatenkov, A., On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold, Kyoto J. Math. 57(4) (2017), 789–806.CrossRefGoogle Scholar
Shioda, T. and Katsura, T., On Fermat varieties, Tohoku Math. J. 31 (1979), 97–115.CrossRefGoogle Scholar
Vial, Ch, Remarks on motives of abelian type, Tohoku Math. J. 69 (2017), 195–220.CrossRefGoogle Scholar
Vial, Ch, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. LMS 106(2) (2013), 410–444.Google Scholar
Vial, Ch, On the motive of some hyperkähler varieties, J. Reine Angew. Math. 725 (2017), 235–247.Google Scholar
Voisin, C., On the Chow ring of certain algebraic hyper-Kähler manifolds, Pure Appl. Math. Q. 4(3) (2008), 613–649.CrossRefGoogle Scholar
Voisin, C., Chow rings and decomposition theorems for K3 surfaces and Calabi–Yau hypersurfaces, Geom. Topol. 16 (2012), 433–473.Google Scholar
Voisin, C., Chow rings, decomposition of the diagonal, and the topology of families (Princeton University Press, 2014).CrossRefGoogle Scholar
Voisin, C., Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties, in K3 Surfaces and Their Moduli, Proceedings of the Schiermonnikoog Conference 2014 (Faber, C., Farkas, G., and van der Geer, G., editors), Progress in Maths 315 (2016).CrossRefGoogle Scholar