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Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

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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 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.

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Acknowledgements

The authors thank Domingo Toledo for describing how Simpson’s work leads to Theorem 1.5 and Olivier Biquard for correspondence about the nonuniform case. We thank Bruno Klingler for explaining the relationship to his conjecture discussed in the introduction, Ian Agol for pointing out Theorem 1.8 and Venkataramana, Nicolas Bergeron, and Emmanuel Ullmo for insightful conversations. We also thank Jean Lécureux and Beatrice Pozzetti for conversations related to the proof of Theorem 1.3(2). It is a pleasure to thank Baldi and Ullmo for having explained their approach to us and for various conversations around Hodge Theory. We are grateful to the referees for the constructive comments and recommendations which improved the quality of our exposition.

Funding

Bader was supported by the ISF Moked 713510 grant number 2919/19. Fisher was supported by NSF DMS-1906107. Miller was supported by NSF DMS-2005438/2300370. Stover was supported by Grant Number 523197 from the Simons Foundation/SFARI and Grants DMS-1906088, DMS-2203555 from the National Science Foundation.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 234 Herzl Street, Rehovot, 7610001, Israel

    Uri Bader

  2. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Fisher

  3. Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

    Nicholas Miller

  4. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Matthew Stover

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  1. Uri Bader
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Correspondence to David Fisher.

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To Domingo Toledo with admiration on the occasion of his 75th birthday.

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Bader, U., Fisher, D., Miller, N. et al. Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds. Invent. math. 233, 169–222 (2023). https://doi.org/10.1007/s00222-023-01186-5

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