SpringerLink
Log in

On the Chow groups of certain cubic fourfolds

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. André, Y.: Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan, …), Séminaire Bourbaki 2003/2004, Astérisque 299 Exp. 929, viii, 115–145

  2. Beauville, A.: Some remarks on Kähler manifolds with c 1 = 0. In: Classification of Algebraic and Analytic Manifolds (Katata, 1982), Birkhäuser Boston, Boston, 1983

    Google Scholar 

  3. Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom., 18(4), 755–782 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math., 301(14), 703–706 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Bloch, S.: Lectures on Algebraic Cycles, Duke Univ. Press, Durham, 1980

    MATH  Google Scholar 

  6. Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Ecole Norm. Sup., 4, 181–202 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Amer. J. Math., 105(5), 1235–1253 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Camere, C.: Symplectic involutions of holomorphic symplectic fourfolds. Bull. Lond. Math. Soc., 44(4), 687–702 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charles, F., Markman, E.: The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces. Comp. Math., 149(3), 481–494 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Corti, A., Hanamura, M.: Motivic decomposition and intersection Chow groups, I. Duke Math. J., 103, 459–522 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delorme, C.: Espaces projectifs anisotropes. Bull. Soc. Math. France, 103, 203–223 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. reine u. angew. Math., 422, 201–219 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Fu, L.: On the action of symplectic automorphisms on the CH0-groups of some hyper-Kähler fourfolds. Math. Z., 280, 307–334 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fu, L.: Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds. Glasgow Math. J., 58(1), 17–37 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gordon, B., Hanamura, M., Murre, J.: Relative Chow—Künneth projectors for modular varieties. J. reine u. angew. Math., 558, 1–14 (2003)

    Article  MATH  Google Scholar 

  16. Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math., 107(3), 447–452 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jannsen, U.: Motivic sheaves and filtrations on Chow groups, in: Motives (U. Jannsen et alii, eds.). Proceedings of Symposia in Pure Math., 55, Part 1 (1994)

  18. 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

    Google Scholar 

  19. Kahn, B., Murre, J., Pedrini, C.: On the transcendental part of the motive of a surface. In: Algebraic Cycles and Motives (J. Nagel and C. Peters, editors), Cambridge University Press, Cambridge, 2007

    Google Scholar 

  20. Kimura, S.: Chow groups are finite dimensional, in some sense. Math. Ann., 331, 173–201 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleiman, S.: Algebraic cycles and the Weil conjectures. In: Dix exposés sur la cohomologie des schémas, North-Holland Amsterdam, 1968, 359–386

    Google Scholar 

  22. Kleiman, S.: The standard conjectures. In: Motives (U. Jannsen et alii, eds.). Proc. Symp. Pure Math., 55, Part 1 (1994)

  23. Laterveer, R.: A remark on the motive of the Fano variety of lines on a cubic. Ann. Math. Québec, to appear

  24. Laterveer, R.: Algebraic cycles on Fano varieties of some cubics, submitted

  25. Laterveer, R.: On Voisin’s conjecture for zero-cycles on hyperkähler varieties, submitted

  26. Murre, J.: On a conjectural filtration on the Chow groups of an algebraic variety. parts I and II. Indag. Math., 4, 177–201 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives. Amer. Math. Soc., University Lecture Series, 61, Providence, 2013

  28. Nagel, J., Saito, M.: Relative Chow—Künneth decompositions for conic bundles and Prym varieties. Int. Math. Res. Not., 16, 2978–3001 (2009)

    MATH  Google Scholar 

  29. Pedrini, C.: On the finite dimensionality of a K3 surface. Manuscripta Mathematica, 138, 59–72 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Scholl, T.: Classical motives, in: Motives (U. Jannsen et alii, eds.). Proc. Symp. Pure Math., 55, Part 1 (1994)

  31. Shen, M., Vial, C.: The Fourier transform for certain hyper-Kähler fourfolds. Memoirs of the AMS, 240(1139), (2016)

  32. Vial, C.: Niveau and coniveau filtrations on cohomology groups and Chow groups. Proc. LMS, 106(2), 410–444 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Vial, C.: Remarks on motives of abelian type. Tohoku Math. J., to appear

  34. Voisin, C.: The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. Ann. Sci. Ecole Norm. Sup., 46(3), 449–475 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Voisin, C.: The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II. J. Math. Sci. Univ. Tokyo, 22, 491–517 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton University Press, Princeton and Oxford, 2014

    Book  MATH  Google Scholar 

  37. Voisin, C.: Hodge structures, Coniveau and Algebraic Cycles, in: “The Legacy of Bernhard Riemann After One Hundred and Fifty Years”, ALM35, Higher Education Press and International Press, Bei**g Boston 2016

    Google Scholar 

Download references

Acknowledgements

Thanks to all participants of the Strasbourg 2014/2015 “groupe de travail” based on the monograph [36] for a very pleasant atmosphere. Thanks to the referees for helpful comments. Many thanks to Kai and Len and Yoyo for stimulating discussions not related to this work.

Author information

Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, CNRS — Université de Strasbourg, 7 Rue René Descartes, 67084, Strasbourg Cedex, France

    Robert Laterveer

Authors
  1. Robert Laterveer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Laterveer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Laterveer, R. On the Chow groups of certain cubic fourfolds. Acta. Math. Sin.-English Ser. 33, 887–898 (2017). https://doi.org/10.1007/s10114-017-6477-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-017-6477-8

Keywords

MR(2010) Subject Classification

Search

Navigation