Projective structures on compex manifolds

我们的主要研究对象是一类紧致复3-流形，它包含一个子域U生物全纯于复射影3-空间中的一个射影线的邻域。我们称这类流形为l类。一般来说，这类流形不承认射影结构，但与承认射影结构的复流形有很深的关系。为了对所有l类流形进行分类，考虑1参数曲面族的奇异纤维附近的Cauduchon度规的存在性问题似乎是非常重要的，我们现在非常接近于完全解。为了实现我们对L类流形的分类计划，S. Ivashkovich关于亚纯映射扩展的一系列定理发挥了重要作用。另一方面，L类流形为我们提供了许多关于亚纯映射可拓问题的有趣例子。N. Okada （Sophia university研究生班）构造了一个Schottky型L类流形的例子，给出了一个Hartogs域到不能在分形集上亚纯扩展的流形的全纯映射，并估计了奇异集的Hausdorff维数。利用Y. Kamozawa (NTT)，我们得到了一个允许全纯投影连接的全纯叶上的特征形式的显式公式。

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.