权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Development of Geometric Complex Analysis

几何复分析的发展

基本信息

批准号：
12440035
负责人：
OHSAWA Takeo
金额：
$ 9.79万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440035/
关键词：
Pseudoconvox domain ∂^^- equation L^2 holomorphic function Levi flatness L^2 division theorem Adherent complex curve Bergman kernel Hefer decomjposition ∂^^-方程式 L^2正則関数の拡張

项目摘要

In spite of a lot of work in complex analysis and conplex analytic geometry in the last century, there seem to remain unnoticed important questions in the basic theory of several complex variables. Relation between the extension and the sivision problems is very likely one of them. The purpose of this research project was to get a new viewpoint of relating extension and division problems on complex manifolds after the author's previous works on the L^2extention theorems for holomorphic functions. As a result we succeeded in improving a well known L^2division theory of H.Skoda. On the other hand, there was a progress concering Levi flat hypersurfaces : Let X be a complex manifold of dimension n and let M be a real hypersurface of X. M is called Levi flat if it locally separates X into two Stein domains i.e.if M is locally psendoconvex from both sides. In recent works of Lins-Nets and Ohsawa, it was proved that complex projective space of dimension n contains no compact real analytic Levi flat hypersurfaces if n 【greater than or equal】 2. It was extended in a joint work with K.Matsumoto by studying the geometry of Levi flat hypersurfaces in complex tori. Unlike the case of projective spaces, tori contain infinitely many compact Levi flat hypersurfaces. We determined all such jypersurfaces under the assumption of real analyticity when the dimension of tori is 2.

尽管上个世纪在复分析和复解析几何方面做了大量的工作，但在多复变基本理论中似乎仍有一些未被注意的重要问题。可拓学与规划的关系问题很可能就是其中之一。本研究的目的是在全纯函数的L^2扩张定理的基础上，对复流形上的扩张与除问题的关系提出一个新的观点。结果，我们成功地改进了著名的H. Skoda的L^2分裂理论。另一方面，关于Levi平坦超曲面的研究也有了新的进展：设X是n维复流形，M是X的真实的超曲面。M称为Levi平坦的，如果它局部地将X分成两个Stein域，即如果M从两侧局部伪凸。在Lins-Nets和Ohsawa最近的工作中，证明了当n [大于或等于] 2时，n维复射影空间不包含紧的真实的解析Levi平坦超曲面.它是延长在一个联合工作与K.松本通过研究几何列维平坦超曲面在复杂的环面。与射影空间不同的是，环面包含无穷多个紧致Levi平坦超曲面。当环面的维数为2时，我们在真实的解析性假设下确定了所有这样的超曲面。