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Research on non-abelian Hodge structures of algebraic varieties

代数簇的非阿贝尔Hodge结构研究

基本信息

批准号：
11640016
负责人：
YOKOGAWA Koji
金额：
$ 2.18万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

There were the following three purposes for this project.1) Study of the non-abelian Hodge decomposition (or filtration) in the case of positive characteristic.2) Study of the relation between defomation theory of non-commutative schemes and non-abelian Hodge theory.3) Study of non-abelian mixed Hodge structures of algebraic varieties.As for the first purpose, I realized that it would be important to construct the crystellin homotopy theory using n-stacks. If such a theory is constructed, the original problem would be clear and there would be many applications. It would be an extention of the work of N. Katz on Frobenius maps and Hodge filtrations in the abelian cases. I am now trying this next big project. On the second one, after the study of non-abelian schemes I understood that it is more natural to describe the deformation parameter space as an n-stacks. Such description would be useful to study the relationship with non-abelian Hodge structures. On the third one, there was a great development by C. Simpson. He describe the non-abelian mixed Hodge structures as kinds of filtrations on non-abelian cohomology n-stacks. I studied his theory and make some explicit calculations of such filtrations on n-stacks. Moreover I worked with Y. Takeda on the pre-Tango structures on curves. The work is related on the first one. With K. Ohba, I studied on the relation between the study of gerbs (using n-stacks) and the study of cycles of moduli spaces.

本课题的主要目的有三个：1）研究非阿贝尔Hodge分解（或过滤）的性质. 2）研究非交换格式的变形理论与非交换Hodge理论之间的关系. 3）研究代数簇的非交换混合Hodge结构.至于第一个目的，我意识到用n-栈来构造crystellin同伦理论是很重要的。如果这样一个理论被建立起来，那么原来的问题就清楚了，而且会有许多应用。这将是N. Katz关于Frobenius映射和交换情形下的Hodge滤。我正在尝试下一个大项目。在第二个方面，在研究了非阿贝尔方案之后，我明白了将变形参数空间描述为n-堆叠是更自然的。这样的描述将有助于研究与非阿贝尔Hodge结构的关系。在第三个方面，C.辛普森他将非交换混合Hodge结构描述为非交换上同调n-栈上的各种过滤。我研究了他的理论，并对n-栈上的这种过滤进行了一些明确的计算。我和Y一起工作。武田在曲线上的前探戈结构。这项工作与第一项有关。与K. Ohba，我研究了gerbs（使用n-栈）的研究与模空间的循环的研究之间的关系。