Special subvarieties of non-arithmetic ball quotients and Hodge Theory

Abstract

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

Added erratum for Corollary 1.3.3 (its proof is not complete). The other results are not affected