Special subvarieties of non-arithmetic ball quotients and Hodge Theory

DOI10.4007/ANNALS.2023.197.1.3arXiv2005.03524MaRDI QIDQ6340245

Emmanuel Ullmo, Gregorio Baldi

Publication date: 7 May 2020

Abstract: Let $GammasubsetoperatornamePU(1,n)$ be a lattice, and $S_{G}amma$ the associated ball quotient. We prove that, if $S_{G}amma$ contains infinitely many maximal totally geodesic subvarieties, then $Gamma$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_{G}amma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_{G}amma$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.

Mathematics Subject Classification ID

Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables (32H02) Discrete subgroups of Lie groups (22E40) Semialgebraic sets and related spaces (14P10) Modular and Shimura varieties (14G35) Model theory of ordered structures; o-minimality (03C64)

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