Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

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dc.citation.firstpage169
dc.citation.journalTitleInventiones mathematicae
dc.citation.lastpage222
dc.citation.volumeNumber233
dc.contributor.authorBader, Uri
dc.contributor.authorFisher, David
dc.contributor.authorMiller, Nicholas
dc.contributor.authorStover, Matthew
dc.date.accessioned2023-07-18T18:47:25Z
dc.date.available2023-07-18T18:47:25Z
dc.date.issued2023
dc.description.abstractFor 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.
dc.identifier.citationBader, Uri, Fisher, David, Miller, Nicholas, et al.. "Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds." <i>Inventiones mathematicae,</i> 233, (2023) Springer Nature: 169-222. https://doi.org/10.1007/s00222-023-01186-5.
dc.identifier.doihttps://doi.org/10.1007/s00222-023-01186-5
dc.identifier.urihttps://hdl.handle.net/1911/114941
dc.language.isoeng
dc.publisherSpringer Nature
dc.rightsThis work is protected by copyright, and is made available here for research and educational purposes. Permission to reuse, publish, or reproduce the work beyond the bounds of Fair Use or other exemptions to copyright law must be obtained from the copyright holder.
dc.titleArithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
dc.typeJournal article
dc.type.dcmiText
dc.type.publicationpost-print
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