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指标，用来度量Hartogs域上的全纯映射到复流形的可扩性。这个指标对于研究复射影三维空间中“大”域的商流形是有用的。这里，射影三维空间P^3中的一个域是“大”的，如果该域包含射影线。这方面的主要结果是正代数维数的大型整环的紧商的刻画。在这种情况下，P^3中的大畴是致密的，而补偿是非常薄的。在另一个假设下，我们可以证明商流形是P^3流形、布兰查德流形或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，我们还利用了S.Ivashkovich关于全纯映射到非Kaehler流形上的扩张的一个结果。2.上述结果是复1维Klein群理论中初等型群情形的类比，并提出了构造高维复解析Klein群理论的可能性。我们可以给出基本的定义来发展三维复解析Klein群理论，并使许多问题公式化。3.我们前期的研究计划是寻找复流形是“可能的”的充分条件。目前，我们在这方面没有任何结果。

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.