The uniqueness and the degeneracy problems of meromorphic maps and the construction of meromorphic maps with deficient devisors.

本课题主要研究亚纯映射的值分布问题，特别是亚纯映射的唯一性问题、退化问题以及亚纯映射的构造问题。从2002年到2003年，项目负责人Aihara和Mori研究了给定超曲面上具有正缺陷的亚纯映射的构造。特别地，我们证明了对于复射影空间上的任意有效因子，存在一个实数区间，使得我们总能构造出缺陷等于区间内每一个值的亚纯映射。这些研究使我们能够更迅速地改进对缺陷的估计。我们还确定了不能评估缺陷的除数。Aihara还考虑了复欧几里德空间的解析覆盖空间与复射影空间之间的亚纯映射的唯一性问题。他特别讨论了某些几何条件对唯一性的约束。从这些作品中，我们实现了项目的第一个目标。我们把项目推进到一些相关的问题上。2004年，亚纯映射域的概化主要由项目负责人Atsuji研究。我们证明了亚纯映射值分布理论的主要工具Nevanlinna理论对于一般完全Kaehler流形上的亚纯函数是可以建立的。在其他成员的努力下，我们在2002-2004年也取得了以下成果：Kitagawa考虑了一个猜想：3球中被封闭曲面包围的区域的体积在封闭曲面的任何等距变形下都是不变的。当封闭面为平环面时，他得到了这个猜想的肯定答案。Kamada证明了承认常数标量曲率的Kaehler度量（可能是不定的）的Hirzebruch曲面必须是投影线的直积的生物全纯曲面。

Our project was concerned with the value distribution of meromorphic maps,in particular, uniqueness problem, degeneration and the construction of meromorphic maps with their defect to arbitrary divisors. From 2002-2003 the project leader Aihara and Mori studied the construction of meromorphic maps with positive defect on given hypersurfaces. In particular, we showed that for an arbitrary effective divisor on complex projective spaces there exists an interval of real numbers such that we can always construct the meromorphic maps whose defect equals to each value in the interval. These researches enabled us to improve the estimate on the defect more sharply. We also determined the divisors which defects cannot be evaluated. Aihara also considered some uniqueness problem of meromorphic maps between analytic covering spaces of complex Euclidean spaces and complex projective spaces. He specially discussed which kinds of constraints to the uniqueness are caused by some geometric conditions. From these works we achieved the first aim of our project. We pushed our project forward to some related problems. In 2004, the generalization of the domain of meromorphic maps was considered mainly by the project leader Atsuji. We showed that Nevanlinna theory, the main tool of the theory of value distribution of meromorphic maps, can be established for meromorphic functions on general complete Kaehler manifolds.We also obtained the following results from 2002-2004 by the effort of the other members.Kitagawa considered a conjecture : the volume of the domain surrounded by a closed surface in 3-sphere is invariant under any isometric deformations of the closed surface. He obtained the affirmative answer to the conjecture in the case when the closed surface is a flat torus.Kamada showed that Hirzebruch surfaces which admit Kaehler metrics (may be indefinite) of constant scalar curvature, must be biholomorphic to a direct product of projective lines.