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Archiv der Mathematik

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

Holomorphic curves in Shimura varieties

  • Michele GiacominiEmail author
Open Access
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

14G35 03C64 14P10 32H02 32M15 

Notes

Acknowledgements

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.

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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

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