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Mathematische Annalen

, Volume 371, Issue 3–4, pp 1229–1253 | Cite as

Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms

  • Colleen Robles
Article

Abstract

Sheng and Zuo’s characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross’s canonical variations of Hodge structure of Calabi–Yau type over (Hermitian symmetric) tube domains.

References

  1. 1.
    Čap, A., Slovák, J.: Parabolic geometries. I. Background and general theory. Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence (2009)Google Scholar
  2. 2.
    Dynkin, E.B.: The maximal subgroups of the classical groups. Am. Math. Soc. Trans 6, 245–378 (1957). Translation of [2]zbMATHGoogle Scholar
  3. 3.
    Gerkmann, R., Sheng, M., van Straten, D., Zuo, K.: On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}}^3\). Math. Ann. 355(1), 187–214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Green, M., Griffiths, P., Kerr, M.: Néron models and boundary components for degenerations of Hodge structure of mirror quintic type. In: Curves and abelian varieties, Contemp. Math., vol. 465, pp. 71–145. Amer. Math. Soc., Providence (2008)Google Scholar
  5. 5.
    Green, M., Griffiths, P., Kerr, M.: Mumford-Tate groups and domains: their geometry and arithmetic. Annals of Mathematics Studies, vol. 183. Princeton University Press, Princeton (2012)Google Scholar
  6. 6.
    Griffiths, P.A.: Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties. Am. J. Math. 90, 568–626 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Griffiths, P.A.: Periods of integrals on algebraic manifolds. II. Local study of the period mapping. Am. J. Math. 90, 805–865 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gross, B.H.: A remark on tube domains. Math. Res. Lett. 1(1), 1–9 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hwang, J.-M., Yamaguchi, K.: Characterization of Hermitian symmetric spaces by fundamental forms. Duke Math. J. 120(3), 621–634 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ivey, T.A., Landsberg, J.M.: Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, vol. 61. American Mathematical Society, Providence (2003)Google Scholar
  11. 11.
    Kostant, B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. Math. 2(74), 329–387 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Landsberg, J.M., Robles, C.: Fubini–Griffiths–Harris rigidity and Lie algebra cohomology. Asian J. Math. 16(4), 561–586 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Matsumoto, K., Sasaki, T., Yoshida, M.: The monodromy of the period map of a 4-parameter family of \(K3\) surfaces and the hypergeometric function of type (3,6). Int. J. Math. 3(1), 164 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sasaki, T., Yamaguchi, K., Yoshida, M.: On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces. In: CR-geometry and overdetermined systems (Osaka, 1994), Adv. Stud. Pure Math., vol. 25, pp. 318–354. Math. Soc. Japan, Tokyo (1997)Google Scholar
  15. 15.
    Sheng, M., Jinxing, X., Zuo, K.: The monodromy groups of Dolgachev’s CY moduli spaces are Zariski dense. Adv. Math. 272, 699–742 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sheng, M., Zuo, K.: Polarized variation of Hodge structures of Calabi–Yau type and characteristic subvarieties over bounded symmetric domains. Math. Ann. 348(1), 211–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentDuke UniversityDurhamUSA

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