Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for manifolds. As in the case, primary result superrigidity theorem certain representations lattices. The proof requires developing new general tools not needed case. Our main results also have number other applications. example, nonexistence maps between manifolds, which related to question Siu, 3-manifolds cannot be and arithmeticity manifolds detected purely by topology underlying variety, Margulis. provide some evidence conjecture Klingler broad generalization Zilber–Pink conjecture.