Algebraic structures and complex structures of Quasi-Abelian varieties

1. 我们利用周期矩阵的标准形式证明了任意n维的q型和0型的拟阿贝尔变换在q维的阿贝尔变换上是C^<*n-q>-主束。本文的证明见：拟阿贝尔变的周期矩阵，九州工业大学学报36(1999),283-286.2。进一步证明了对于每一个q型和s型的拟阿贝尔变量C^n/Γ，存在一个q型和0型的拟阿贝尔变量C^n/Γ，满足以下条件：(1)C^n/Γ是C^n/Γ_0的覆盖流形；(2)C^n/Γ_0导出了C^n/Γ上定义拟阿贝尔结构的充足黎曼形式。结合1的结果，得到了A.Andreotti-F得到的拟阿贝尔变分的纤化定理。盖拉德利和y . abe。上述结果已在第三届实复分析国际研讨会（6月，韩国梨花女子大学）、日本数学学会秋季会议（9月，京都大学）和第四届实复分析国际研讨会（10月，广岛大学）上公布。包括这些结果在内的论文发表在第三届实与复分析国际研讨会论文集（2000），75-82和第四届实与复分析国际研讨会论文集（2000），81-87 .4。由2的结果，我们可以由相关的0类拟阿贝尔变元构造每一个拟阿贝尔变元。我们有很多例子稍后会公布，使用的是由科学研究资助基金带来的计算机和软件。

1. We proved that any n-dimensional Quasi-Abelian variety of type q and kind 0 is a C^<*n-q>-principal bundle over an Abelian variety of dimension q, using our standard form of the period matrix. The proof is found in the paper : Period Matrices of Quasi-Abelian Varieties, Bulletin of the Faculty of Engineering, Kyushu Sangyo University 36(1999), 283-286.2. Further we showed that for every Quasi-Abelian variety C^n/Γ of type q and kind s, there exists an associated Quasi-Abelian variety C^n/Γ of type q and kind 0 satisfying the following conditions :(1) C^n/Γ is a covering manifold of C^n/Γ_0 and(2) The ample Riemann form which defines a Quasi-Abelian structure on C^n/Γ is induced by C^n/Γ_0. Combining this and the result of 1, we get the fibration theorems of Quasi-Abelian varieties which was obtained by A.Andreotti-F.Gherardelli and Y.Abe.3. The above results were announced in the Third International Workshop on Real and Complex Analysis (June, Ewha Womans University, Korea), Autumn Conference of Mathematical Society of Japan (September, Kyoto University) and the Fourth International Workshop on Real and Complex Analysis (October, Hiroshima University). Papers including these results were published in the Proceedings of the Third International Workshop on Real and Complex Analysis (2000), 75-82 and the Proceedings of the Fourth International Workshop on Real and Complex Analysis (2000), 81-87 .4. From the result of 2, we can construct every Quasi-Abelian variety from the associated Quasi-Abelian variety of kind 0. We have many examples of them which will be published later, using the computers and the softwares which was brought by the Grant-in-Aid for Scientic Research.