复双曲流形的算术性、超刚性和全测地子流形
Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
摘要：
对于$n \ge 2$，我们证明了一个有限体积复双曲$n$ -流形包含无穷多个最大适当浸入全测地子流形的尺寸至少为2是算术，平行于我们以前的工作，真正的双曲流形。在实双曲情形下，我们的主要结果是复双曲格的某些表示的超刚性定理。证明需要开发新的通用工具不需要在真正的双曲的情况下。我们的主要结果也有一些其他的应用。例如，我们证明了复双曲流形之间某些映射的不存在性，这与Siu的一个问题有关，某些双曲$3$ -流形不可能是复双曲流形的全测地子流形，复双曲流形的算术性纯粹是由底层复簇的拓扑来检测的，这与Margulis的一个问题有关.我们的结果也提供了一些证据Klingler猜想是一个广泛的推广Zilber-Pink猜想。
英文摘要：
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.
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