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

1.我们在以前的工作中定义了一个度量Hartogs域上的全纯映射到复流形的可扩性的指标。这个指标对于研究复射影3-空间中“大”整环的商流形是有用的。这里，射影3-空间P^3中的一个域是“大”的，如果该域包含投影线。这方面的主要结果是刻画了具有正代数维大整环的紧商。在这种情况下，P^3中的大区域是稠密的，并且比非常薄。在另一个假设下，我们可以证明商流形是P^3、Blanchard流形或L-Hopf流形。我们认为，这一额外的假设很快就可以被取消。为了得到这一结果，我们还利用S.IVASKOVICH关于将全纯映射推广到非Kaehler流形的最新结果。2.上述结果类似于一维复Klein群论中的初等型群情形，并暗示了构造高维复解析Klein群论的可能性。我们可以给出发展三维复解析Klein群理论的基本定义，并提出许多问题。3.我们早期的研究计划是寻找复流形是“可能”的好的充分条件。目前，我们在这个方向上还没有结果。

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.