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Extension property of holomorphic maps and complex structures of the target manifolds

全纯映射的可拓性质和目标流形的复杂结构

基本信息

批准号：
17540094
负责人：
KATO Masahide
金额：
$ 1.22万
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：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

1.We defined in our previous work an index which measures the extendibility of holomorphic maps of a Hartogs domain to complex manifolds. This index is useful to study quotient manifolds of "large" domains in complex projective 3-spaces. Here a domain in projective 3-space P^3 is "large", if the domain contains projective lines. Main result in this direction is the characterization of compact quotient of large domains with positive algebraic dimension. In this case, the large domain is dense in P^3 and the compement is very thin. Under an additional assumption, we could show that the quotient manifold is a P^3, a Blanchard manifold, or a L-Hopf manifold. We think that this additional assumption can be removed soon. To obtain this result we use also a recent result of S.Ivashkovich on extension of holomorphic maps into non-Kaehler manifolds.2.The result above is an analogy of the elementary type group case in Klein group theory of complex 1-dimension and suggests a possibility of constructing higher dimensional complex analytic Klein group theory. We could give basic definitions to develop 3-dimensional complex analytic Klein group theory and formulate many problems.3.Our research plan in the early stage was to seek good sufficient conditions of complex manifolds to be "probable". At present we have no results in this direction.

1.在之前的工作中，我们定义了一个度量Hartogs域的全纯映射到复流形的可扩展性的指标。该指标对复射影3-空间中“大”域商流形的研究是有用的。这里，射影3空间P^3中的一个域是“大”的，如果该域包含射影线。这一方向的主要成果是具有正代数维数的大域的紧商的刻画。在这种情况下，大的定义域在P^3上是密集的，而补很薄。在一个附加的假设下，我们可以证明商流形是一个P^3，一个Blanchard流形，或一个L-Hopf流形。我们认为这一额外的假设可以很快消除。为了得到这个结果，我们还使用了s.i avashkovich关于全纯映射扩展到非kaehler流形的最新结果。上述结果类比了复一维Klein群论中的初等型群情况，为构造高维复解析Klein群论提供了可能。我们可以给出基本的定义来发展三维复解析克莱因群论并形式化许多问题。我们前期的研究计划是寻求复杂流形“可能”的良好充分条件。目前我们在这方面还没有结果。