Tautological ring of strata of differentials



Strata of k-differentials on smooth curves parameterize sections of the k-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the \(\kappa \) and \(\psi \) classes of moduli spaces of pointed smooth curves along with the class \(\eta = \mathcal O(-1)\) of the Hodge bundle. We show that if there is no pole of order k, then the tautological ring is generated by \(\eta \) only, and otherwise it is generated by the \(\psi \) classes corresponding to the poles of order k.

Mathematics Subject Classification

14H10 14H15 14C25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbarello, E., Cornalba, M.: Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Publ. Math. Inst. Hautes Études Sci. 88, 97–127 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., Möller, M.: Compactification of strata of Abelian differentials. Duke Math. J. (to appear) Google Scholar
  3. 3.
    Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., Möller, M.: Strata of \(k\)-differentials. Algebr. Geom. (to appear) Google Scholar
  4. 4.
    Boissy, C.: Connected components of the moduli space of meromorphic differentials. Comment. Math. Helv. 90(2), 255–286 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, D.: Strata of abelian differentials and the Teichmüller dynamics. J. Mod. Dyn. 7(1), 135–152 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, D.: Teichmüller dynamics in the eyes of an algebraic geometer. Proc. Symp. Pure Math. 95, 171–197 (2017)Google Scholar
  7. 7.
    Chen, D.: Affine geometry of strata of differentials. J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748017000445
  8. 8.
    Edidin, D.: The codimension-two homology of the moduli space of stable curves is algebraic. Duke Math. J. 67(3), 241–272 (1992)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eskin, A., Kontsevich, M., Zorich, A.: Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ. Math. Inst. Hautes Études Sci. 120, 207–333 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eskin, A., Mirzakhani, M.: Invariant and stationary measures for the SL \((2,\mathbb{R})\) action on Moduli space. Publ. Math. Inst. Hautes Études Sci. (to appear) Google Scholar
  11. 11.
    Eskin, A., Mirzakhani, M., Mohammadi, A.: Isolation, equidistribution, and orbit closures for the SL \((2,\mathbb{R})\) action on moduli space. Ann. Math. (2) 182(2), 673–721 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. Aspects Math. E 33, 109–129 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Faber, C., Pandharipande, R.: Logarithmic series and Hodge integrals in the tautological ring. Mich. Math. J. 48, 215–252 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Farkas, G., Pandharipande, R.: The moduli space of twisted canonical divisors, with an appendix by F. Janda, R. Pandharipande, A. Pixton, D. Zvonkine. J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748016000128
  15. 15.
    Filip, S.: Splitting mixed Hodge structures over affine invariant manifolds. Ann. Math. (2) 183, 681–713 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130(1), 1–37 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3), 631–678 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Korotkin, D., Zograf, P.: Tau function and moduli of differentials. Math. Res. Lett. 18(3), 447–458 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lanneau, E.: Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 1–56 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Looijenga, E.: On the tautological ring of \(\cal{M}_g\). Invent. Math. 121(2), 411–419 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sauvaget, A.: Cohomology classes of strata of differentials. arXiv:1701.07867
  22. 22.
    Wright, A.: Translation surfaces and their orbit closures: an introduction for a broad audience. EMS Surv. Math. Sci. 2, 63–108 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zorich, A.: Flat surfaces. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics, and Geometry I, pp. 437–583. Springer, Berlin (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA

Personalised recommendations