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Complex analysis on quasi-Abelian varieties

拟阿贝尔簇的复分析

基本信息

批准号：
13640199
负责人：
UMENO Takashi
金额：
$ 2.24万
依托单位：
Kyushu Sangyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640199/
关键词：
quasi-Abelian variety Abelian variety period matrix toroidal group de Rham cohomology ^^-__∂-cohomology quasi-Abelian variety Abelian variety toroidal group complex torus periodic matrix cohomology Quasi-Abelian Variety Toroidal

项目摘要

1. We obtained all standard forms of period matrices for quasi-Abelian varieties. Using these forms, we got a brief proof of fibration theorems for quasi-Abelian varieties. Further, we have an another proof, which does not depend on the theory of harmonic integrals, of classical Riemann conditions for Abelian varieties.2. The author gave a talk about the above results at the Fifth International Workshop on Real and Complex Analysis (October 2001, Hiroshima. University). He wrote the paper : Period matrices for quasi-Abelian Varieties, which includes de Rham cohomology of toroidal groups, period matrices and fibration theorems for quasi-Abelian varieties. This will be published in Japanese Journal of Mathematics vol.29-1(2003).3. From the theorems of period matrices, we found the existence of quasi-Abelian varieties which have principal bundle structures over non-algebraic complex tori. Further we could construct many examples of toroidal groups which have no non-constant meromorphic fu … More nctions on them. This is an extension of a method of Siegel who constructed examples of complex tori which have no non-constant meromorphic functions on them. We could not construct these examples without the computers and the softwares which was brought by the Grant-in-Aid for Scientic Research. Because we needed an enormous amount of calculations of matrices including polynomials.4. The author gave an lecture about the theory of period matrices as an invited talk at the conference of the mathematical society of Japan, March 2002. Further, he gave a talk about these examples obtained by the theorems of period matrices at the Sixth International Workshop on Real and Complex Analysis (December 2002, Hiroshima University). By these results, We think that the main aims of the research was obtained.5. As above, we use computers as important tools for a study of mathematics. The method of using computers for mathematics which we have developed will be applicable for mathematical educations. The author gave a talk about a computerization of mathematical education at the conference of information processing education (October 2002, Tokyo University). We think these methods were also obtained by the research. Less

1. 我们得到了拟阿贝尔变量周期矩阵的所有标准形式。利用这些形式，我们得到了拟阿贝尔变分的纤化定理的一个简短证明。更进一步，我们得到了另一个不依赖于调和积分理论的关于阿贝尔变项的经典黎曼条件的证明。作者在第五届实复分析国际研讨会（2001年10月，广岛）上发表了上述成果。大学)。撰写了论文《拟阿贝尔变体的周期矩阵》，其中包括环面群的de Rham上同调、周期矩阵和拟阿贝尔变体的纤化定理。这将发表在日本数学杂志第29卷第1期（2003）。从周期矩阵的定理出发，证明了非代数复环面上具有主束结构的拟阿贝尔变型的存在性。更进一步，我们可以构造出许多环面群上没有非常亚纯函数的例子。这是西格尔构造没有非常亚纯函数的复环面实例方法的推广。如果没有科学研究补助金带来的计算机和软件，我们就无法构建这些例子。因为我们需要大量的矩阵计算，包括多项式。作者于2002年3月应邀在日本数学学会会议上作了关于周期矩阵理论的报告。此外，他在2002年12月广岛大学第六届实复分析国际研讨会上发表了关于周期矩阵定理得到的这些例子的演讲。通过这些结果，我们认为达到了研究的主要目的。如上所述，我们使用计算机作为学习数学的重要工具。我们开发的利用计算机进行数学运算的方法将适用于数学教育。作者在2002年10月东京大学信息处理教育学术会议上作了关于数学教育计算机化的演讲。我们认为这些方法也是通过研究得出的。少