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Algebraic cycles and intersections of three quadrics

Part of: Cycles and subschemes

Published online by Cambridge University Press:  07 September 2021

ROBERT LATERVEER
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

Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 349 - 367
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

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

References

Addington, N.. The derived category of the intersection of four quadrics, arXiv:0904.1764Google Scholar
Beauville, A.. Prym varieties and the Schottky problem. Invent. Math. 41 (1977), 149—196.Google Scholar
Beauville, A.. Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273 (1986), 647—651.Google Scholar
Beauville, A.. Determinantal hypersurfaces, Michigan Math. J. 48 no. 1 (2000), 39—64.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).10.1017/CBO9781107325968.004CrossRefGoogle Scholar
Beauville, A. and Voisin, C.. On the Chow ring of a K3 surface, J. Alg. Geom. 13 (2004), 417—426.10.1090/S1056-3911-04-00341-8CrossRefGoogle Scholar
Bergeron, N. and Li, Z.. Tautological classes on moduli space of hyperkähler manifolds, Duke Math. J. 168 no. 7 (2019), 1179—1230.10.1215/00127094-2018-0063CrossRefGoogle Scholar
Bloch, S. and Srinivas, V.. Remarks on correspondences and algebraic cycles. Amer. J. Math. Vol. 105 no. 5 (1983), 1235—1253.10.2307/2374341CrossRefGoogle Scholar
Bondal, A. and Orlov, D.. Derived categories of coherent sheaves, in: Proc. Internat. Congress of Mathematicians, Vol. II (Beijing, 2002), 47—56, (Higher Ed. Press, Beijing 2002).Google Scholar
Delorme, P.. Espaces projectifs anisotropes. Bull. Soc. Math. France 103 no. 2 (1975), 203—223.10.24033/bsmf.1802CrossRefGoogle Scholar
Diaz, H.. The Chow ring of a cubic hypersurface, to appear in Internat. Math. Res. Notices.Google Scholar
Dixon, A.. Note on the reduction of a ternary quantic to a symmetric determinant. Proc. Cambridge Phil. Soc. 11 (1902), 350—351.Google Scholar
Fu, L.. Laterveer, R. and Vial, Ch.. The generalised Franchetta conjecture for some hyper-Kähler varieties (with an appendix joint with M. Shen), J. Math. Pures et Appliquées (9) 130 (2019), 1—35.10.1016/j.matpur.2019.01.018CrossRefGoogle Scholar
Fu, L.. Laterveer, R. and Vial, Ch.. The generalised Franchetta conjecture for some hyper-Kähler varieties, II, Journal de l’Ecole Polytechnique–Mathématiques 8 (2021), 1065—1097.10.5802/jep.166CrossRefGoogle Scholar
Fu, L.. Laterveer, R. and Vial, Ch.. Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type. Annali Mat. Pura ed Applicata 200 no. 5 (2021), 2085—2126.Google Scholar
Fu, L.. Tian, Z. and Vial, Ch.. Motivic hyperkähler resolution conjecture for generalised Kummer varieties. Geometry and Topology 23 (2019), 427—492.Google Scholar
Fu, L. and Vial, Ch.. Distinguished cycles on varieties with motive of abelian type and the section property. J. Alg. Geom. 29 (2020), 53—107.Google Scholar
Fulton, W.. Intersection theory. (Springer–Verlag Ergebnisse der Mathematik, Berlin Heidelberg New York Tokyo 1984.)Google Scholar
Green, M. and Griffiths, P.. An interesting 0-cycle, Duke Math. J. 119 no. 2 (2003), 261—313.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, Cambridge 2007).10.1017/CBO9781107325968.008CrossRefGoogle Scholar
Kimura, S.-I.. Chow groups are finite dimensional, in some sense. Math. Ann. 331 no. 1 (2005), 173—201.Google Scholar
Kuznetsov, A.. Derived categories of quadric fibrations and intersections of quadrics. Adva. Math. 218 no. 5 (2008), 1340—1369.Google Scholar
Laszlo, Y.. Théorème de Torelli générique pour les intersections complètes de trois quadriques de dimension paire. Invent. Math. 98 (1989), 247—264.Google Scholar
Laterveer, R.. A remark on the Chow ring of Küchle fourfolds of type d3. Bull. Aust. Math. Soc. 100 no. 3 (2019), 410—418.Google Scholar
Laterveer, R. and Vial, Ch.. On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder varieties. Canad. J. Math. 72 no. 2 (2020), 505—536.Google Scholar
Laterveer, R.. Algebraic cycles and Verra fourfolds. Tohoku Math. J. 72 no. 3 (2020), 451—485.Google Scholar
Laterveer, R.. On the Chow ring of certain Fano fourfolds. Ann. Univ. Paedagog. Crac. Stud. Math. 19 (2020), 39—52.Google Scholar
Laterveer, R.. On the Chow ring of Fano varieties of type S2. Abh. Math. Semin. Univ. Hambg. 90 (2020), 17—28.10.1007/s12188-020-00218-8CrossRefGoogle Scholar
Laterveer, R.. Algebraic cycles and Fano threefolds of genus 8, preprint.Google Scholar
Laterveer, R.. Algebraic cycles and intersections of two quadrics. Mediterranean J. Math. (2021), doi: 10.1007/s00009-021-01787-5.CrossRefGoogle Scholar
Mukai, S.. Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77 (1984), 101—116.Google 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.Google Scholar
Murre, J.. Nagel, J. and Peters, C.. Lectures on the theory of pure motives. Amer. Math. Soc. University Lecture Series 61 (Providence 2013).10.1090/ulect/061CrossRefGoogle Scholar
Negut, A.. Oberdieck, G. and Yin, Q.. Motivic decompositions for the Hilbert scheme of points of a K3 surface, arXiv:1912.09320v1.Google Scholar
O’Grady, K.. The Hodge structure of the intersection of three quadrics in an odd-dimensional projective space. Math. Ann. 273 (1986), 277—285.Google Scholar
O’Grady, K.. Decomposable cycles and Noether–Lefschetz loci, Doc. Math. 21 (2016), 661—687.Google Scholar
Otwinowska, A.. Remarques sur les cycles de petite dimension de certaines intersections complètes, C. R. Acad. Sci. Paris 329 (1999), Série I, 141—146.10.1016/S0764-4442(99)80478-9CrossRefGoogle Scholar
Pavic, N.. Shen, J. and Yin, Q.. On O’Grady’s generalized Franchetta conjecture. Int. Math. Res. Notices (2016), 1—13.10.1093/imrn/rnw158CrossRefGoogle Scholar
Scholl, T.. Classical motives, in: Motives (U. Jannsen et alii, eds.). Proc. Symposia in Pure Math. vol. 55 (1994), Part 1.10.1090/pspum/055.1/1265529CrossRefGoogle Scholar
Shen, M. and Vial, Ch.. The Fourier transform for certain hyperKähler fourfolds. Mem. Amer. Math. Soc. 240 (2016), no. 1139.Google Scholar
Shen, M. and Vial, Ch.. The motive of the Hilbert cube $X^{[3]}$. Forum Math. Sigma 4 (2016), 55 pp..10.1017/fms.2016.25CrossRefGoogle Scholar
Tavakol, M.. The tautological ring of the moduli space $M_2^{rt}$. Internat. Math. Res. Notices 178 (2013).10.1093/imrn/rnt178CrossRefGoogle Scholar
Tavakol, M.. Tautological classes on the moduli space of hyperelliptic curves with rational tails. J. Pure Applied Algebra 222 no. 8 (2018), 2040—2062.Google Scholar
Terasoma, T.. Complete intersections of hypersurfaces—the Fermat case and the quadric case. Japan J. Math. (N.S.) 14 no. 2 (1988), 309—384.Google Scholar
Vial, Ch.. Niveau and coniveau filtrations on cohomology groups and Chow groups. Proc. Landon Math. Soc. 106(2) (2013), 410—444.10.1112/plms/pds031CrossRefGoogle Scholar
Vial, Ch.. On the motive of some hyperkähler varieties. J. Reine Angew. Math. 725 (2017), 235—247.Google Scholar
Voisin, C.. Remarks on filtrations on Chow groups and the Bloch conjecture. Annali Mat. Pura ed Applicada 183 no. 3 (2004), 421—438.Google Scholar
Voisin, C.. Chow Rings, Decomposition of the Diagonal, and the Topology of Families. (Princeton University Press, Princeton and Oxford, 2014).10.1515/9781400850532CrossRefGoogle Scholar
Yin, Q.. The generic nontriviality of the Faber–Pandharipande cycle. Internat. Math. Res. Notices 2015 no. 5 (2015), 1263—1277.Google Scholar
Yin, Q.. Finite-dimensionality and cycles on powers of K3 surfaces. Comment. Math. Helv. 90 (2015), 503–511.Google Scholar