Geometriae Dedicata

, Volume 194, Issue 1, pp 1–28 | Cite as

Tautological rings on Jacobian varieties of curves with automorphisms

  • Thomas RichezEmail author
Original Paper Area 1


Let J be the Jacobian of a smooth projective complex curve C which admits non-trivial automorphisms, and let \(\mathrm{{A}}(J)\) be the ring of algebraic cycles on J with rational coefficients modulo algebraic equivalence. We present new tautological rings in \(\mathrm{{A}}(J)\) which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683–688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of J.


Algebraic cycles Tautological rings Jacobians Automorphisms Fourier transforms 

Mathematics Subject Classification (2010)

14C15 14C25 14H37 14H40 



This article is part of my thesis written at the University of Strasbourg. I would like to thank my Ph.D. advisor Professor Rutger Noot for his continual support and numerous suggestions which contributed to the improvement of this paper. I also want to thank the referee for his comments which helped me to reduce the length of this paper.


  1. 1.
    Arap, M.: Algebraic cycles on Prym varieties. Math. Ann. 353(3), 707–726 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of algebraic curves. Vol. I, vol. 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1985)Google Scholar
  3. 3.
    Beauville, A.: Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne. In: Algebraic Geometry (Tokyo, Kyoto, 1982), Lecture Notes in Math., vol. 1016, pp. 238–260. Springer, Berlin (1983)Google Scholar
  4. 4.
    Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273(4), 647–651 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beauville, A.: Algebraic cycles on Jacobian varieties. Compos. Math. 140(3), 683–688 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Birkenhake, C., Lange, H.: Complex abelian varieties, second ed., vol. 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2004)Google Scholar
  7. 7.
    Colombo, E., van Geemen, B.: Note on curves in a Jacobian. Compos. Math. 88(3), 333–353 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Drezet, J.-M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97(1), 53–94 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fulton, W.: Intersection theory, second ed., vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (1998)Google Scholar
  10. 10.
    Milne, J.S.: Jacobian varieties. In: Arithmetic Geometry (Storrs, Conn., 1984), pp. 167–212 . Springer. New York (1986)Google Scholar
  11. 11.
    Mumford, D.: Abelian varieties, vol. 5 of Tata Institute of Fundamental Research Studies in Mathematics. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second edition (1974)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.CNRS, IRMA UMR 7501Université de StrasbourgStrasbourgFrance

Personalised recommendations