Complex Line Bundles on Toroidal Groups and the Spaces of Holomorphic Sections

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1. We described the conditions for a toroidal group to be a Quasi-Abelian variety using a Z-valued skew-symmetric form. We got this result by studying the relations between the complex structures and the rational structures of a toroidal group, calculating de Rham cohomology groups and *-cohomology groups of a toroidal group through a Fourier analytic method. In a classical case, the so-called Riemann conditions for a complex torus to be an Abelian variety were established by the theory of harmonic integrals. Our results include these facts.2. From the above results, we got standard forms of the period matrices of Quasi- Abelian varieties. Applying these results to the compact cases, we got also the standard forms of the period matrices of Abelian Varieties.3. Using the results of 2, we see that a Hermitian form which defines a Quasi-Abelian variety is obtained by the curvature form of a positive line bundle. Since the space of the sections of the positive line bundle defines an embedding of a Quasi-Abelian variety, the purpose of our study has been almost obtained. We do not know a concrete description of the space of the sections which defines an embedding, comparing the case of an Abelian variety. But, since we found the relations between the Quasi-Abelian varieties and Abelian varieties through their period matrices, it will be possible to represent a concrete form of the space of the sections.4. Our results of 1 were published in the proceedings at the first congress of ISAAC in the university of Delaware in U.S.A and the fifth international conference on complex analysis in Peking University in China, 1997. And the results of 2 were published in the proceedings of the sixth international conference on complex analysis in Andong University in Korea, 1998. Futher, the results of 3 will be publishe in the proceedings at the second congress of ISAAC and the seventh international conference of complex analysis in Japan, 1999.