Holomorphic curves in Shimura varieties

TL;DR

This paper extends the Bloch-Ochiai theorem to non-compact Shimura varieties, establishing the Zariski closure properties of holomorphic curves and completing the proof for all Shimura varieties.

Contribution

It provides a hyperbolic analogue of the Bloch-Ochiai theorem for all Shimura varieties, including the non-compact case, building on prior work for compact varieties.

Findings

Proved the hyperbolic analogue for non-compact Shimura varieties.

Completed the proof of the theorem for all Shimura varieties.

Extended the understanding of holomorphic curves in Shimura varieties.

Abstract

We prove an 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 analysed was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.