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

1. 我们用z值偏对称的形式描述了环面群是拟阿贝尔变的条件。通过研究环面群的复结构与有理结构之间的关系，利用傅里叶解析方法计算环面群的de Rham上同群和*-上同群，得到了这一结果。在经典的情况下，所谓的黎曼条件的复环是一个阿贝尔变化是由调和积分理论建立的。我们的结果包括这些事实。由上述结果，我们得到了拟阿贝尔变量周期矩阵的标准形式。将这些结果应用于紧情况，得到了阿贝尔变周期矩阵的标准形式。利用2的结果，我们看到定义拟阿贝尔变化的厄米形式是由正线束的曲率形式得到的。由于正线束的截面空间定义了拟阿贝尔变量的嵌入，因此我们的研究目的基本得到了满足。我们不知道定义嵌入的部分空间的具体描述，比较阿贝尔变化的情况。但是，由于我们通过它们的周期矩阵发现了拟阿贝尔变体和阿贝尔变体之间的关系，因此将有可能表示截面空间的具体形式。我们的结果发表在美国特拉华大学ISAAC第一届大会和中国北京大学第五届复分析国际会议论文集上。结果发表在1998年韩国安东大学第六届国际复合体分析会议论文集上。此外，3的结果将发表在1999年在日本举行的第二届ISAAC大会和第七届国际复杂分析会议的论文集上。

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.