Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

8907582这个项目的目标是考虑复杂的微分几何中的几个问题。它们包括分析定义在几个复变量空间中的域上的函数和映射，以及研究域及其边界的几何。要做的工作将包括确定一族紧致复流形的极限是否又是这种类型的流形。紧致复流形都是同一不可约厄米对称流形的双全纯映象。最近已经推出了解决这个问题的新技术。此外，还对复射影空间的情形给出了完整的证明。这项工作-紧致型不可约Hermitan对称流形的整体不可变形性是强刚性理论中未解决的部分之一的弱化版本。该理论的目的是证明某些紧致复流形可以用拓扑条件或曲率条件来刻画。第二个研究方向是关于全纯函数的值分布的高维Nevanlinna理论的持续研究。等维情形已经得到了高度发展，并得到了相当好的理解。在这项工作中，将考虑非等维项。以前的方法在此设置中不起作用，可能必须使用全纯连接的新方法。这一领域的结果与数论有着直接的联系。此外，我们还将研究有界全纯函数的构造。在某种意义上，经典的均匀化定理回答了一维中的任何这样的问题，因为任何单连通的黎曼曲面要么等价于球面，要么等价于平面或圆盘。关于紧致负曲Kahler流形的泛覆盖空间，同样的分类问题一直没有得到解决。等价地，人们会问覆盖空间是否允许大量的有界全纯函数。除了少数特殊情况外，没有人产生过任何这样的函数(除了常量)。Kohn-Fefferman关于全纯函数空间投影的Holder估计的最新结果表明，他们的方法有可能被移植到流形上，并导致一种构造期望函数的技术。如果这成为现实，在更高维度上攻击日冕问题的可能性将最终实现。

8907582 Siu The object of this project is to consider several problems in complex differential geometry. They include analysis of functions and mappings defined on domains in the space of several complex variables as well as studies of the geometry of the domains and their boundaries. Work to be done will include an effort to determine whether the limit of a family of compact complex manifolds which are all biholomorphic images of the same irreducible Hermitian symmetric manifold is again a manifold of this type. New techniques for attacking this problem have been introduced recently. In addition, a complete proof for the case of complex projective space has been givng. This work - the global nondeformability of irreducible Hermitan syymetric manifolds of compact type is a weakened version of one of the unresolved parts of the theory of strong rigidity. The purpose of the theory is to show that certain compact complex manifolds can be characterized either by topological or curvature conditons. A second line of investigation concerns continuing research on the higher dimensional Nevanlinna theory of value distribution of holomorphic functions. The equidimensional case has been highly developed and is reasonably well understood. In this work, the nonequidimensional will be considered. Past methods do not work in this setting and newer methods of holomorphic connections will probably have to be used instead. Results in this area have direct connections with number theory. Additional work will be done on the construction of bounded holomorphic functions. In one sense, the classical uniformization theorem answers any such question in one dimension since any simply connected Riemann surface is either equivalent to the sphere, the plane or a disc. The same classification question concerning the universal covering space for compact negatively curved Kahler manifolds has never been resolved. Equivalently, one asks whether the covering space admits a large number of bounded holomorphic functions. Except for a few special cases, no one has produced any such functions (other than constants). The recent results of Kohn-Fefferman on Holder estimates of projections into spaces of holomorphic functions holds out the possibility that their method may be transplanted to manifolds and lead to a technique for construction of the desired functions. If that becomes a reality, the likelihood of an attack on the Corona problem in higher dimensions would finally be realized.