Research on Complex Analytic Properties of Complex Homogeneous Spaces

本课题的主要目的之一是寻找李群齐次空间具有（双-）不变CR结构的充分必要条件。接下来我们的任务是对它们进行复分析分类。在项目期间，我们得到了CR向量空间范畴的一些基本性质。我们可以应用它们来调查上述条件，然后进行分类。然而，这些任务目前还没有完成，而是留给未来的研究。我们现在列出了每位研究者的一些显著成果。(1) Takeuchi的结果：他给出了CR向量空间的CR张量积的新概念，推广了模的张量积的概念。(2) Hatada研究了正特征模形式空间及其子空间在结构上的关系。他还研究了复模形式空间及其不变子空间中Hecke算子轨迹的显式计算方法及其应用。(3) Fujimoto的结果：(i)一个具有多个扭转截面的有理椭圆曲面，通过双元变换，转化为具有多个纤维的有理椭圆曲面。（ii）具有非负Kodaira维的非奇异射影三联体，允许一个（非同构）到自全纯，具有非奇异极小模型，当Kodaira维=2或0时，它具有与阿贝尔三联体同构的有限覆盖，或光滑曲面与椭圆曲线的直接积。

One of the main objectives of this research project is to find necessary and sufficient conditions for homogeneous spaces of Lie groups to have a (bi-)invariant CR structure. Then next we have the task to classify them complex analytically. During the term of the project we obtained some of the basic properties of the category of CR vector spaces. We could apply them to investigate the conditions mentioned above and then to pursue the classification. These tasks are, however, not yet accomplished at the present moment, but are left for the future study We now list up some of the remarkable results of each investigator.(1) Takeuchi's result : He gave a new notion of CR tensor product of CR vector spaces, which generalizes that of the tensor product in the category of modules.(2) Hatada studied relations, on structure, among spaces of modular forms in positive characteristics and theirsubspaces. He studied also methods for explicit computations of the traces of Hecke operators with respect to spaces of complex modular forms and their invariant subspaces, and applications of them.(3) Fujimoto's result : (i) A rational elliptic surface with many torsion sections is, by a birational transformation, transformed to a rational elliptic surface with a multiple fiber. (ii) A non-singular projective threefold with a non-negative Kodaira dimension admitting an (nonisomorphic) onto self-holomorphism, has a non-singular minimal model and if the Kodaira dimension=2, or 0, it has a finite etale covering isomorphic to an Abelian threefold, or a direct product of a smooth surface and an elliptic curve.