Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

8907582 兆 这个项目的目的是考虑几个问题 复杂的微分几何中。 其中包括分析 定义在域上的函数和映射 几个复杂的变量，以及研究的几何形状， 域及其边界。 要做的工作将包括努力确定 紧致复流形族的极限 它们都是同一不可约 厄米对称流形也是这种类型的流形。 已经引入了解决这个问题的新技术 最近 此外，还对以下情形给出了完整的证明： 复射影空间的定义。 这项工作- 不可约Hermitan对称的整体不可变形性 紧型流形是其中一个流形的弱化形式， 强刚性理论中未解决的部分。 目的 理论的一个重要方面是证明某些紧致复流形 可以用拓扑或曲率来表征 条件。 第二条调查线涉及继续 高维Nevanlinna价值理论研究 全纯函数的分布 等维的 案件已经高度发达，并相当不错 明白 在这项工作中，非等维将是 考虑了 过去的方法在这种情况下不起作用， 全纯联络的方法可能必须 使用代替。 这一领域的成果与 数字理论。 还将在以下方面开展额外工作： 全纯函数 从某种意义上说， 单值化定理回答任何这样的问题在一个 维数，因为任何单连通黎曼曲面都是 相当于球体、平面或圆盘。 相同的 关于泛覆盖空间的分类问题 对于紧致负曲Kahler流形， 解决了 等价地，有人问覆盖空间是否 允许大量有界全纯函数。 除了 对于少数特殊情况，没有人产生任何这样的函数 （除了常数）。 Kohn-Fergusman最近的研究结果 关于全纯空间投影的保持器估计 函数提供了一种可能性，即它们的方法可能 移植到流形上，并导致一种技术， 构建所需的功能。 如果这成为一个 事实上，在2011年， 更高的维度将最终实现。

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.