Skip to main content
SpringerLink
Log in

On the Chow group of zero-cycles of Calabi–Yau hypersurfaces

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We prove the existence of a canonical zero-cycle \(c_X\) on a Calabi–Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, \(n=\mathrm {dim}\,X\) is proportional to the class of a point on a rational curve in X.

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

Access this article

Log in via an institution

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. Beauville, A.: On the splitting of the Bloch–Beilinson filtration. In: Algebraic Cycles and Motives (vol. 2), London Mathematical Society Lecture Notes 344, pp. 38–53 (2007)

  2. Beauville, A., Voisin, C.: On the Chow ring of a K3 surface. J. Algebr. Geom. 13, 417–426 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brion, M.: Lectures on the geometry of flag varieties (2004). arXiv:math/0410240

  4. Brion, M., Kumar, S.: Frobenius Splitting Methods in Geometry and Representation Theory, volume 231 of Progress in Mathematics. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  5. Eisenbud, D., Harris, J.: 3264 and all that: A second course in algebraic geometry, pp. 632. Cambridge University Press (2016) (ISBN-10:1107602726)

  6. Fu, L.: Beauville–Voisin conjecture for generalized Kummer varieties. Int. Math. Res. Not. (2014)

  7. Huybrechts, D.: Curves and cycles on K3 surface. Algebr. Geom. 1, 69–106 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Knop, F., Kraft, H., Vust, T.: The Picard group of a G-variety in Algebraische Transformationsgruppen und Invariantentheorie. DMV Seminar, vol. 13. Birkhäuser Verlag, Basel (1989)

  9. Lin, H.Y.: On the Chow group of zero-cycles of a generalized Kummer variety (2015). arXiv:1507.05155v1

  10. Mumford, D.: Rational equivalence of zero-cycles in surfaces. J. Math. Kyoto Univ. 9, 195–204 (1968)

    MATH  Google Scholar 

  11. Perrin, N.: Courbes rationnelles sur les variétés homogenes. Ann. Inst. Fourier 52(1), 105–132 (2002)

    Article  MathSciNet  Google Scholar 

  12. Roitman, A.: Rational equivalence of zero-dimensional cycles. Mat. Sb. (N.S.) 89(131), 569–585 (1972)

    MathSciNet  Google Scholar 

  13. Roitman, A.: The torsion of the group of zero-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)

    Article  MathSciNet  Google Scholar 

  14. Voisin, C.: On the Chow ring of certain algebraic hyper-Kähler manifolds. Pure Appl. Math. Q. 4(3), 613–649 (2008)

    Article  MATH  Google Scholar 

  15. Voisin, C.: Rational equivalence of zero-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady. In: Recent Advances in Algebraic Geometry, A Conference in Honor of Rob Lazarsfeld’s 60th Birthday (to appear) (2012)

  16. Voisin, C.: Remarks and questions on coisotropic subvarieties and zero-cycles of hyper-Kähler varieties (2015). arXiv:1501.02984

  17. Welters, G. E.: Abel–Jacobi isogenies for certain types of Fano threefolds. Mathematical Centre Tracts, vol. 141, pp. 139. Mathematisch Centrum, Amsterdam (1981) (ISBN: 90-6196-227-7)

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Paris, France

    Ivan Bazhov

Authors
  1. Ivan Bazhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Bazhov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bazhov, I. On the Chow group of zero-cycles of Calabi–Yau hypersurfaces. manuscripta math. 152, 189–197 (2017). https://doi.org/10.1007/s00229-016-0853-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-016-0853-z

Mathematics Subject Classification

Search

Navigation