Family of curves with large unitary summand in the Hodge bundle



In this note, we construct non-isotrivial families of curves of genus \(g\ge 2\), where the rank of the unitary summand contained in the Hodge bundle can be as large as \((2g+1)/3\), and hence disprove Xiao’s conjecture for the unitary rank.


Family Unitary bundle Xiao’s conjecture 

Mathematics Subject Classification

14D06 14H10 14D99 



The author would like to thank K. Zuo for many useful discussion. He is also grateful to L. Stoppino and V. González-Alonso for a careful reading of a draft of this note.


  1. 1.
    Albano, A., Pirola, G.P.: Dihedral monodromy and Xiao fibrations. Ann. Mat. Pura Appl. (4) 195(4), 1255–1268 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arakelov, S.J.: Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat. 35, 1269–1293 (1971)MathSciNetGoogle Scholar
  3. 3.
    Barja, M.Á., González-Alonso, V., Naranjo, J.C.: Xiao’s conjecture for general fibred surfaces. J. Reine Angew. Math. (2014). arXiv:1401.7502
  4. 4.
    Barja, M.Á., Stoppino, L.: Linear stability of projected canonical curves with applications to the slope of fibred surfaces. J. Math. Soc. Japan 60(1), 171–192 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barja, M.Á., Zucconi, F.: On the slope of fibred surfaces. Nagoya Math. J. 164, 103–131 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cai, J.-X.: Irregularity of certain algebraic fiber spaces. Manuscr. Math. 95(3), 273–287 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Catanese, F., Dettweiler, M.: The direct image of the relative dualizing sheaf needs not be semiample. C. R. Math. Acad. Sci. Paris 352(3), 241–244 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Catanese, F., Dettweiler, M.: Vector bundles on curves coming from variation of Hodge structures. Int. J. Math. 27(7), 1640001, 25 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, K., Xin, L., Zuo, K.: On the Oort conjecture for Shimura varieties of unitary and orthogonal types. Compos. Math. 152(5), 889–917 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Debarre, O.: Inégalités numériques pour les surfaces de type général. Bull. Soc. Math. France 110(3), 319–346 (1982). (With an appendix by A. Beauville) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deligne, P.: Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progr. Math., vol. 67, pp. 1–19. Birkhäuser Boston, Boston (1987)Google Scholar
  12. 12.
    Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Japan 30(4), 779–794 (1978)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fujita, T.: The sheaf of relative canonical forms of a Kähler fiber space over a curve. Proc. Japan Acad. Ser. A Math. Sci. 54(7), 183–184 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    González-Alonso, V., Stoppino, L., Torelli, S.: On the rank of the flat unitary factor of the hodge bundle (2017). arXiv:1709.05670v1
  15. 15.
    Kollár, J.: Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, pp. 361–398. North-Holland, Amsterdam (1987)Google Scholar
  16. 16.
    Lu, X., Zuo, K.: On the slope conjecture of Barja and Stoppino for fibred surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5).
  17. 17.
    Lu, X., Zuo, K.: On the slope of hyperelliptic fibrations with positive relative irregularity. Trans. Am. Math. Soc. 369(2), 909–934 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Moonen, B., Oort, F.: The Torelli locus and special subvarieties, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, pp. 549–594. Int. Press, Somerville (2013)Google Scholar
  19. 19.
    Pirola, G.P.: On a conjecture of Xiao. J. Reine Angew. Math. 431, 75–89 (1992)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Serrano, F.: Isotrivial fibred surfaces. Ann. Mat. Pura Appl. (4) 171, 63–81 (1996)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Simpson, C.T.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3(3), 713–770 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tan, S.-L.: On the invariants of base changes of pencils of curves. I. Manuscr. Math. 84(3–4), 225–244 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276(3), 449–466 (1987)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xiao, G.: Irregularity of surfaces with a linear pencil. Duke Math. J. 55(3), 597–602 (1987)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xiao, G.: Irregular families of hyperelliptic curves, Algebraic geometry and algebraic number theory (Tianjin, 1989–1990), Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 3, pp. 152–156. World Sci. Publ., River Edge (1992)Google Scholar
  26. 26.
    Zuo, K.: On the negativity of kernels of Kodaira–Spencer maps on Hodge bundles and applications. Asian J. Math. 4(1), 279–301 (2000). Kodaira’s issueMathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiPeople’s Republic of China

Personalised recommendations