Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

TL;DR

This paper proves that complex hyperbolic manifolds with infinitely many certain geodesic submanifolds are arithmetic, establishing superrigidity results and exploring implications for maps, topology, and conjectures in complex hyperbolic geometry.

Contribution

It introduces new superrigidity theorems for complex hyperbolic lattices and develops tools that extend previous real hyperbolic results to the complex setting.

Findings

Proves arithmeticity of certain complex hyperbolic manifolds with many geodesic submanifolds

Establishes nonexistence of specific maps between complex hyperbolic manifolds

Provides evidence supporting a conjecture related to the Zilber--Pink conjecture

Abstract

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