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A Brief Note Concerning Hard Lefschetz for Chow Groups

Published online by Cambridge University Press:  20 November 2018

Robert Laterveer
Robert Laterveer*
Affiliation:
Institut de Recherche Mathématique Avancée, Université de Strasbourg, 6 Rue René Descartes, 67084 Strasbourg CEDEX, France e-mail: laterv@math.unistra.fr
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Abstract

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We formulate a conjectural hard Lefschetz property for Chow groups and prove it in some special cases, roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griõths group. An appendix includes related statements that follow from results of Vial.

Keywords

14C1514C2514C30algebraic cyclesChow groupsfinite-dimensional motives

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016, pp. 144 - 158
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] André, Y., Motifs de dimension finie (d'après S.-I. Kimura, P. O'Sullivan …). Séminaire Bourbaki, 2003/2004, Astérisque 299(2005), no. 929, viii, 115–145.Google Scholar
[2] André, Y., Motivation for Hodge cycles. Adv. Math. 207(2006), no. 2, 762–781. http://dx.doi.Org/10.1016/j.aim.2006.01.005 Google Scholar
[3] Bloch, S. and Ogus, A.. Gersten's conjecture and the homology of schemes. Ann. Sci. École Norm. Sup. (4) 7(1974), 181–201.Google Scholar
[4] Bloch, S. and Srinivas, V.. Remarks on correspondences and algebraic cycles. Amer. J. Math. 105,(1983), no. 5,1235-1253. http://dx.doi.Org/10.2307/2374341 Google Scholar
[5] Fu, B., Remarks on hard Lefschetz conjectures on Chow groups. Sci. China Math. 53(2010), 105–114. http://dx.doi.Org/10.1007/s11425-009-021 8-0 Google Scholar
[6] Guletskiï, V. and Pedrini, C.. The Chow motive of the Godeaux surface. In: Algebraic geometry, de Gruyter, Berlin, 2002.Google Scholar
[7] Hartshorne, R., Equivalence relations on algebraic cycles and subvarieties of small codimension. In: Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), American Mathematical Society, Providence, RI, 1975 Google Scholar
[8] N, J. N.. Iyer, Murre's conjectures and explicit Chow-Kunneth projectors for varieties with a nef tangent bundle. Trans. Amer. Math. Soc. 361(2009), 1667–1681. http://dx.doi.Org/1 0.1 090/S0002-9947-08-04582-0 Google Scholar
[9] Jannsen, U., Mixed motives and algebraic K-theory. Lecture Notes in Mathematics, 1400, Springer-Verlag, Berlin, 1990.Google Scholar
[10] Jannsen, U., Motivic sheaves and filtrations on Chow groups. In: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, American Mathematical Society, Providence, RI, 1994.Google Scholar
[11] Jannsen, U., Equivalence relations on algebraic cycles. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer, Dordrecht, 2000, pp. 225–260.Google Scholar
[12] Kimura, S.-I., Chow groups are finite dimensional, in some sense. Math. Ann. 331(2005), no. 1, 173–201, http://dx.doi.Org/10.1007/s00208-004-0577-3 Google Scholar
[13] Kleiman, S. L., Algebraic cycles and the Weil conjectures. In: Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 359–386.Google Scholar
[14] Kleiman, S. L., The standard conjectures. In: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, American Mathematical Society, Providence, RI, 1994, pp. 3–20.Google Scholar
[15] Laterveer, R., Algebraic varieties with small Chow groups. J. Math. Kyoto Univ. 38(1998), no. 4, 673–694.Google Scholar
[16] Mumford, D., Rational equivalence ofO-cycles on surfaces. J. Math. Kyoto Univ. 9(1969), no. 2, 195–204.Google Scholar
[17] Murre, J., Nagel, J.. and Peters, C.. Lectures on the theory of pure motives. University Lecture Series, 61, American Mathematical Societym, Providence, RI, 2013.Google Scholar
[18] Pedrini, C., On the finite dimensionality of a K3 surface. Manuscripta Math. 138(2012), no. 1-2, 59–72. http://dx.doi.Org/1 0.1 007/s00229-011 -0483-4 Google Scholar
[19] Vial, C., Algebraic cycles andfibrations. Doc. Math. 18(2013), 1521–1553.Google Scholar
[20] Vial, C., Projectors on the intermediate algebraic Jacobians. New York J. Math. 19(2013), 793–822.Google Scholar
[21] Vial, C., Remarks on motives ofabelian type. arxiv:111 2.1 080Google Scholar
[22] Vial, C., Niveau and coniveau filtrations on cohomology groups and Chow groups. Proc. London Math. Soc. 106(2013), 410–444. http://dx.doi.Org/1 0.1112/plms/pdsO31 Google Scholar
[23] Vial, C., Chow-Kiinneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles. J. Algebraic Geom. 24(2015), 51–80. http://dx.doi.Org/10.1090/S1056-3911-2014-00616-0 Google Scholar
[24] Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Notices 4(1995), 187–198.Google Scholar
[25] Voisin, C., Remarks on zero-cycles on self-products of varieties. In: Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996, pp. 265–285.Google Scholar
[26] Voisin, C., The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. Ann. Sci. Éc. Norm. Super 46(2013), 449–475.Google Scholar
[27] Voisin, C., Chow rings, decomposition of the diagonal, and the topology of families. Annals of Mathematics Studies, 187, Princeton University Press, Princeton, NJ, 2014.Google Scholar