Abstract
Let $\Gamma \subset \mathrm{PU}(1,n)$ be a lattice and $S_\Gamma$ be the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove an Ax--Schanuel Conjecture for $S_\Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_\Gamma$ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.
Citation
Gregorio Baldi. Emmanuel Ullmo. "Special subvarieties of non-arithmetic ball quotients and Hodge theory." Ann. of Math. (2) 197 (1) 159 - 220, January 2023. https://doi.org/10.4007/annals.2023.197.1.3
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