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Projective structures on compex manifolds

复流形上的射影结构

基本信息

批准号：
09640129
负责人：
KATO Masahide
金额：
$ 2.18万
依托单位：
Sophia University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640129/
关键词：
complex manifold projective structure Gauduchon metric Hartogs domain non-Kahler extension peoblem Weyl curvature teusor Chern forms Characteristic forms projective connections 正則写像 Gauduchonメトリック特異ファイバー正則拡張性

项目摘要

Our main object of study is a class of compact complex 3-manifolds which contain a subdomain U biholomorphic to a neighborhood of a projective line ill a complex projective 3-space. We call such a class of manifolds by Class L. Such manifolds do not admit projective structures in general, but very deeply related to complex manifolds which admit projective structures. To classify all Class L. manifolds, it seems very important to consider the existence problem of Cauduchon metric near a singular fibre of a 1-parameter family of surfaces, and we are now very near to the complete solution.To carry out our plan of classification of Class L manifolds, a series of theorems of S. Ivashkovich on the extension of meromorphic maps play important roles. On the other hand, our manifolds of Class L supply us with many interesting examples on extension problems of meromorphic maps. N. Okada (Sophia Univ. Graduate Course) constructed an example of Schottky type Class L manifolds, gave a holomorphic map of a Hartogs domain to the manifold whichl cannot he extended meromorphically across a fractal set, and estimated the Hausdorff dimension of the singular set. With Y. Kamozawa (NTT), we have obtained a explicit formula of characteristic forms on a holomorphic foliation which admits a holomorphic projective connection.

本文的主要研究对象是一类紧致复三维流形，它包含一个与复射影三维空间中的射影线的邻域双全纯的子域U。我们把这样一类流形称为L类。这样的流形一般不允许投射结构，但与允许投射结构的复流形有很深的关系。对所有L类进行分类。在L类流形的分类中，考虑1-参数曲面族的奇异纤维附近的Cauduchon度量的存在性问题是非常重要的，我们现在已经非常接近于完全解. Ivashkovich对亚纯映射的扩张起着重要的作用。另一方面，我们的L类流形为我们提供了许多有趣的例子亚纯映射的扩张问题。N. Okada（Sophia Univ. Graduate Course）构造了一个Schottky型L类流形的例子，给出了Hartogs域到不能在分形集上亚纯扩张的流形的全纯映射，并估计了奇异集的Hausdorff维数.与Y。Kamozawa（NTT）的方法，得到了容许全纯射影联络的全纯叶理的特征形的一个显式公式。