Archiv der Mathematik

, Volume 111, Issue 4, pp 379–388 | Cite as

Holomorphic curves in Shimura varieties

  • Michele GiacominiEmail author
Open Access


We prove a hyperbolic analogue of the Bloch–Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.


Holomorphic curve Shimura varieties O-minimality Pila–Wilkie Bloch–Ochiai 

Mathematics Subject Classification

14G35 03C64 14P10 32H02 32M15 



I would like to thank my supervisor Andrei Yafaev for pointing me to this problem and some helpful discussions. I would also like to thank Jacob Tsimerman who was the first to point out that the result in Lemma 3.3 could be generalised from the cocompact case in [12] to the general case using the existence of a bounded realisation for \(\mathcal{D}\). Finally, I would like to thank Chirstopher Daw, Ziyang Gao, Jacob Tsimerman, and the referee for helpful comments on an earlier version of the paper.

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.


  1. 1.
    Ash, A., Mumford D., Rapoport, M., Tai, Y.: Smooth compactifications of locally symmetric varieties. 2nd edn. Cambridge Mathematical Library. With the collaboration of Peter Scholze, pp. x+230. Cambridge University Press, Cambridge (2010)Google Scholar
  2. 2.
    Hwang, J.-M., To, W.-K.: Volumes of complex analytic subvarieties of Hermitian symmetric spaces. Am. J. Math. 124(6), 1221–1246 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Klingler, B., Ullmo, E., Yafaev, A.: The hyperbolic Ax–Lindemann–Weierstrass conjecture. Publ. Math. Inst. Hautes Études Sci. 123, 333–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren der Mathematischen Wissenschaften, vol. 318. Springer, Berlin (1998)Google Scholar
  5. 5.
    Mok, N.: Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds. Series in Pure Mathematics, vol. 6. World Scientific Publishing Co., Inc., Teaneck, NJ (1989)Google Scholar
  6. 6.
    Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math. 42, 239–272 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Pila, J., Tsimerman, J.: The André–Oort conjecture for the moduli space of abelian surfaces. Compos. Math. 149(2), 204–216 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pila, J., Wilkie, A.J.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pila, J.: O-minimality and the André–Oort conjecture for \(\mathbb{C}^{n}\). Ann. Math. (2) 173(3), 1779–1840 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ullmo, E.: Applications du théorème d’Ax–Lindemann hyperbolique. Compositio. Math. 150(2), 175–190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ullmo, E., Yafaev, A.: Hyperbolic Ax–Lindemann theorem in the cocompact case. Duke Math. J. 163(2), 433–463 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ullmo, E., Yafaev, A.: Holomorphic curves in compact Shimura varieties (2016). arXiv:1610.01494 [math.AG]
  13. 13.
    Ullmo, E., Yafaev, A.: O-minimal flows on abelian varieties. Q. J. Math. 68(2), 359–367 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    van den Dries, L.: Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series 248, p. x+180. Cambridge University Press (1998)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University College LondonLondonUK

Personalised recommendations