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Holomorphic curves in Shimura varieties. (English) Zbl 1408.14090

Arch. Math. 111, No. 4, 379-388 (2018); correction ibid. 114, No. 1, 119-121 (2020).
Summary: 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.

MSC:

14G35 Modular and Shimura varieties
03C64 Model theory of ordered structures; o-minimality
14P10 Semialgebraic sets and related spaces
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:

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