ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 63, Issue 2, pp 315–321 | Cite as

Some elementary examples of quartics with finite-dimensional motive

Article

Abstract

This small note contains some easy examples of quartic hypersurfaces that have finite-dimensional motive. As an illustration, we verify a conjecture of Voevodsky (concerning smash-equivalence) for some of these special quartics.

Keywords

Algebraic cycles Chow groups Motives Finite-dimensional motives Quartics 

Mathematics Subject Classification

14C15 14C25 14C30 

Notes

Acknowledgments

This note is a belated echo of the Strasbourg 2014–2015 groupe de travail based on the monograph [23]. Thanks to all the participants for the pleasant and stimulating atmosphere. Many thanks to Yasuyo, Kai and Len for lots of enjoyable post-work apéritifs.

References

  1. 1.
    André, Y.: Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan,...), Séminaire Bourbaki 2003/2004, Astérisque 299 Exp. No. 929, viii, 115—145 (2005)Google Scholar
  2. 2.
    Brion, M.: Log homogeneous varieties. In: Actas del XVI Coloquio Latinoamericano de Algebra, Revista Matemática Iberoamericana, Madrid (2007). arXiv: math/0609669
  3. 3.
    de Cataldo, M., Migliorini, L.: The Chow groups and the motive of the Hilbert scheme of points on a surface. J. Algebra 251(2), 824–848 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Deligne, P.: La conjecture de Weil pour les surfaces \(K3\). Invent. Math. 15, 206–226 (1972)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Guletskiĭ, V., Pedrini, C.: The Chow motive of the Godeaux surface. In: Beltrametti, M.C., et al. (eds.) Algebraic Geometry, a volume in memory of Paolo Francia. Walter de Gruyter, Berlin, New York (2002)Google Scholar
  6. 6.
    Ivorra, F.: Finite dimensional motives and applications (following S.-I. Kimura, P. O’Sullivan and others). In: Autour des motifs, Asian-French summer school on algebraic geometry and number theory, Volume III. Panoramas et synthèses, Société mathématique de France (2011)Google Scholar
  7. 7.
    Iyer, J.: Murre’s conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle. Trans. Am. Math. Soc. 361, 1667–1681 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Iyer, J.: Absolute Chow-Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties. Mich. Math. J. 60(1), 79–91 (2011)CrossRefMATHGoogle Scholar
  9. 9.
    Jannsen, U.: On finite–dimensional motives and Murre’s conjecture. In: Nagel, J., Peters, C. (eds.) Algebraic cycles and motives. Cambridge University Press, Cambridge (2007)Google Scholar
  10. 10.
    Kahn, B., Sebastian, R.: Smash-nilpotent cycles on abelian 3-folds. Math. Res. Lett. 16, 1007–1010 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Katsura, T., Shioda, T.: On Fermat varieties. Tohoku Math. J. 31(1), 97–115 (1979)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kimura, S.: Chow groups are finite dimensional, in some sense. Math. Ann. 331, 173–201 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives. Amer. Math. Soc. University Lecture Series, vol. 61, Providence (2013)Google Scholar
  14. 14.
    Pedrini, C., Weibel, C.: Some surfaces of general type for which Bloch’s conjecture holds. In: Period Domains, Algebraic Cycles, and Arithmetic. Cambridge Univ. Press, Cambridge (2015)Google Scholar
  15. 15.
    Pedrini, C.: On the finite dimensionality of a \(K3\) surface. Manuscr. Math. 138, 59–72 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Shioda, T.: The Hodge conjecture for Fermat varieties. Math. Ann. 245, 175–184 (1979)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    van Geemen, B.: Kuga–Satake varieties and the Hodge conjecture. In: Gordon, B., et al. (eds.) The Arithmetic and Geometry of Algebraic Cycles, Banff 1998. Kluwer, Dordrecht (2000)Google Scholar
  18. 18.
    Vial, C.: Remarks on motives of abelian type. Tohoku Math. J. arXiv:1112.1080
  19. 19.
    Vial, C.: Projectors on the intermediate algebraic Jacobians. N. Y. J. Math. 19, 793–822 (2013)MathSciNetMATHGoogle Scholar
  20. 20.
    Vial, C.: Chow-Künneth decomposition for \(3\)- and \(4\)-folds fibred by varieties with trivial Chow group of zero-cycles. J. Algebr. Geom. 24, 51–80 (2015)CrossRefMATHGoogle Scholar
  21. 21.
    Voevodsky, V.: A nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Not. 4, 187–198 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Voisin, C.: Bloch’s conjecture for Catanese and Barlow surfaces. J. Differ. Geom. 97, 149–175 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families. Princeton University Press, Princeton and Oxford (2014)CrossRefMATHGoogle Scholar
  24. 24.
    Xu, Z.: Algebraic cycles on a generalized Kummer variety. arXiv:1506.04297v1

Copyright information

© Università degli Studi di Ferrara 2016

Authors and Affiliations

  1. 1.CNRS-IRMA, Université de StrasbourgStrasbourg CedexFrance

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