**11****月2日**

The Hyperbolic Ax-Lindemann-Weierstrass Conjecture for arbitrary lattices in the rank-1 case

**（曹锡华数学论坛）**

讲座题目：The Hyperbolic Ax-Lindemann-Weierstrass Conjecture for arbitrary lattices in the rank-1 case

主讲人：莫毅明 教授

主持人：吴瑞聪

开始时间：2015-11-0213:00-14:30

讲座地址：闵行校区数学楼102报告厅

主办单位：数学系 科技处

报告人简介：莫毅明为香港大学终身教授，亦曾为美国哥伦比亚大学和巴黎十一大学终身教授。莫教授是世界知名的复几何专家，在多复变、复微分几何与代数几何等领域里做出了非常重大的贡献。莫教授毕业于史丹福大学，是1994年世界数学家大会的45分钟报告者。

报告内容简介:

In the theory of Shimura varieties, there is the famous André-Oort Conjecture according to which, given a Shimura variety *X*, the Zariski closure of a set of special points on *X *is necessarily a Shimura subvariety. Since the underlying complex-analytic space of a Shimura variety is a finite-volume quotient of a bounded symmetric domain Ω, the problem has a close connection with the geometry of bounded symmetric domains. In fact, following the work of Pila-Zannier on the Manin-Mumford Conjecture (about torsion points on an algebraic curve canonically embedded in its Jacobian), it has been established by Ullmo that a proof of André-Oort Conjecture can be broken down into two parts. Denoting by π: Ω→*X *the uniformization map of a Shimura variety *X*, where Ω is a bounded symmetric domain in its Harish-Chandra realization, the scheme of proof consists of a purely number-theoretic statement concerning Galois orbits of special points and a purely geometric statement according to which the the Zariski closure of the image π(*S*) of an algebraic subset *S *ofΩ is necessarily totally geodesic in *X*. The latter statement is called the Ax-Lindemann-Weierstrass Conjecture, which has now been established in a series of works of Pila-Tsimmerman (2014), Ullmo-Yafaev (2014) and Klingler-Ullmo-Yafaev (Preprint 2015). Their proofs are based on methods of Model Theory in Mathematical Logic, but the proof of the general case involves a substantial amount of Complex Geometry especially the work of Hwang-To (2002) on volumes of subvarieties of bounded symmetric domains.

For Shimura varieties *X *= Ω */ *Γ, the lattices Γ are by assumption arithmetic. Arithmeticity plays a crucial role in model-theoretic methods in the proof of the Hyperbolic Ax-Lindemann-Weierstrass Conjecture. From the geometric perspective, the latter conjecture should remain valid for arbitrary lattices. In case Ω is of rank 1, i.e., Ω = *B*^{n}is the complex unit ball, starting from a completely geometric problem in 2010 we showed using methods of Kähler geometry that the Zariski closure of the image of a totally geodesic complex submanifold is always totally geodesic without assuming that Γ is arithmetic. The author noticed at the same time that an adaptation of the proof yields a proof of the analogue of the Hyperbolic Ax-Lindemann-Weierstrass Conjecture for arbitrary lattices in the rank-1 case (without being aware of the conjecture). In this lecture we develop the general machinery for arbitrary lattices on bounded symmetric domains and give a complete proof of the aforementioned adaptation in the rank-1 case.