论文标题
复杂双曲线歧管的算术，超级繁殖和完全的测量亚策略
Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
论文作者
论文摘要
对于$ n \ ge 2 $，我们证明有限体积复杂双曲线$ n $ - 包含无限的许多最大浸入尺寸的完全测量的亚货物，至少两个是算术的，这与我们以前的实际屈光歧管平行。与实际的双曲线情况一样，我们的主要结果是用于复杂双曲线晶格的某些表示的超级疏松定理。证明需要在实际双曲线情况下开发不需要的新通用工具。我们的主要结果还具有许多其他应用程序。例如，我们证明复杂的双曲线歧管之间某些地图不存在，这与SIU的问题有关，这是某些双曲线$ 3 $ - 雄性不能完全是复杂双曲线歧管的地理亚序列，并且复杂多重歧管的算术纯粹是由复杂的多样的，这是相关的，这是复杂的，这是一个与之相关的复杂杂物。我们的结果还为克林格勒的猜想提供了一些证据，这是对Zilber-粉碎猜想的广泛概括。
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.
