Let \(\Gamma \subset \text{PU} (1, n)\) be a lattice (meaning that \(\Gamma \backslash \text{PU} (1, n)\) admits a finite invariant measure). Such a lattice \(\Gamma\) is said to be arithmetic if it lies in the commensurability class of some image \(p (\mathbf{G} (\mathbb{Z}))\), where \(\mathbf{G}\) is a semisimple linear algebraic group over \(\mathbb{Q}\) and where \(p : \mathbf{G} (\mathbb{R}) \to \text{PU} (1, n)\) is a surjective homomorphism with compact kernel. This article gives a sufficient condition for \(\Gamma\) to be arithmetic in terms of the associated ball quotient \(S_{\Gamma} = \Gamma \backslash X\), where \(X\) is the hermitian symmetric space associated to \(\text{PU} (1, n)\); it does this by showing that \(\Gamma\) is arithmetic if \(S_{\Gamma}\) contains infinitely many maximal complex totally geodesic subvarieties. (The converse does not hold; in fact, there exist arithmetic \(\Gamma\) for which \(S_{\Gamma}\) does not allow any strict totally geodesic subvariety.)
This main result is proved by studying unlikely intersection phenomena inside a period domain for polarised \(\mathbb{Z}\)-variations of Hodge structures. These techniques also yield an alternative proof of the characterization of arithmetic lattices by means of a commensurability criterion, a classical result that is originally due to G. A. Margulis [Discrete subgroups of semisimple Lie groups. Berlin etc.: Springer-Verlag (1991; Zbl 0732.22008)].
Enabling the proof of the main result is the following idea. An argument using infinitesimal rigidity shows that the ball quotient \(S_{\Gamma}\) admits a \(\mathbb{Z}\)-variation of Hodge structures \(\widehat{\mathbb{V}}\). Complex-analytically, this gives rise to a period map \(\psi : S_{\Gamma}^{\text{an}} \to \widehat{\mathbf{G}} (\mathbb{Z}) \backslash D\). This gives rise to two bi-algebraic structures on \(S_{\Gamma}\), and hence to two notions of special (that is, bi-algebraic) subvarieties of \(S_{\Gamma}\). The first of these notions (that of \(\Gamma\)-special subvarieties of \(S_{\Gamma}\)) comes from the universal cover \(\pi : X \to S_{\Gamma}\); these are the images of the algebraic subvarieties of \(X\) along \(\pi\). The second notion (that of \(\mathbb{Z}\)-special subvarieties of \(S_{\Gamma}\)) uses the period map \(\psi\); these are the analytic irreducible components of \(\psi^{-1} (\pi_{\mathbb{Z}} (D'))\) that come from Mumford–Tate subdomains of \(D\). Arguably the main discovery of this paper, one that allows one to relate \(\mathbb{Z}\)-special subvarieties with unlikely intersection, is that the notions of \(\Gamma\)-special and \(\mathbb{Z}\)-special subvarieties of \(S_{\Gamma}\) coincide; both are, in turn, simply those subvarieties of \(S_{\Gamma}\) that are totally geodesic.
The same techniques also allow the authors to apply work by Bakker-Tsimerman on the Ax-Schanuel conjecture [B. Bakker and J. Tsimerman, Invent. Math. 217, No. 1, 77–94 (2019; Zbl 1420.14021)] in the context of the current paper. This yields a proof of the following non-arithmetic version of the Ax-Schanuel conjecture : Let \(W \subset X \times S_{\Gamma}\) be an algebraic subvariety, and let \(\Pi : X \times S_{\Gamma}\) be the graph of the universal cover \(\pi : X \to S_{\Gamma}\). Then if \(U\) is an irreducible component of \(W \cap \Pi\) such that \(\dim (U) > \dim (W) - \dim (S_{\Gamma})\), we have that the projection of \(U\) to \(S_{\Gamma}\) is either zero-dimensional or contained in a strict totally geodesic subvariety of \(S_{\Gamma}\). Another consequence of the results in this article is that if \(H \subset \text{PU} (1, n)\) is of hermitian type, then the intersection \(\Gamma \cap H\) is a lattice in \(H\) provided that it is Zariski dense in \(H\).