Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

• 成果类型: Article
• 署名作者: Bader, Uri; Fisher, David; Miller, Nicholas; Stover, Matthew
• 署名单位: Weizmann Institute of Science; Rice University; University of Oklahoma System; University of Oklahoma - Norman; Pennsylvania Commonwealth System of Higher Education (PCSHE); Temple University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01186-5
• 发表日期: 2023
• 页码: 169-222
• 关键词: maximal representations; lattices; RIGIDITY; bundles; equidistribution; commensurability; subvarieties; conjecture; SPACE

摘要

For n >= 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 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.