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STRUCTURE OF THE MAPPING CLASS GROUP AND GEOMETRY OF THE MODULI SPACE OF RIEMANN SURFACES

黎曼曲面的映射类群结构及模空间几何

基本信息

批准号：
10440016
负责人：
MORITA Shigeyuki
金额：
$ 5.18万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440016/
关键词：
mapping class group Riemann surface moduli space Torelli group family of Riemann surfaces monodromy ジョンソン準同型ジョンソン準同形

项目摘要

We have investigated the structure of the mapping class group of surfaces as well as the geometry of moduli space of Riemann surfaces mainly from the viewpoints of topology. The main results we obtained are as follows.(i) The subalgebra of the rational cohomology algebra of the mapping class group of surfaces generated by the Mumford-Morita classes is called the tautological algebra. There have been three approaches to the study of this tautological algebra. The first is based on the twisted Mumford-Morita classes introduced by Kawazumi, the second is in terms of invariants for trivalent graphs and the third is through symplectic representation theory. Summarizing previous results, we found that the above three approaches correspond exactly to each others.(ii) The theory of secondary characteristic classes of the mapping class group is still a largely unknown area. However, in this research, we proved that these secondary characteristic classes have deep structures that cannot be detected by the nilpotent completion of the Torelli group. It seems highly likely that the solvable or semi-simple structure of the Torelli group will become more and more important in the future.(iii) We made significant progress in understanding the algebro-geometrical as well as the topological structure of families of Riemann surfaces. In particular, we obtained many results concerning the monodromies of symplectic fibrations.

本文主要从拓扑学的角度研究了曲面的映射类群的结构以及黎曼曲面的模空间几何。我们得到的主要结果如下。(i)由Mumford-Morita类生成的曲面的映射类群的有理上同调代数的子代数称为重言式代数。有三种方法来研究这个重言式代数。第一个是基于扭曲的Mumford-Morita类Kawazumi介绍，第二个是在三价图的不变量和第三个是通过辛表示理论。总结以往的结果，我们发现，上述三种方法完全对应于对方。(ii)映射类群的次特征类理论在很大程度上还是一个未知的领域。然而，在这项研究中，我们证明了这些次级特征类具有无法通过Torelli群的幂零完备化检测到的深层结构。Torelli群的可解或半简单结构在未来很可能会变得越来越重要。(iii)我们取得了重大进展，了解代数几何以及拓扑结构的家庭黎曼曲面。特别地，我们得到了许多关于辛纤维化的monodromies的结果。