Research on Complex Analytic Properties of Complex Homogeneous Spaces

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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.