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Geometry and topology of 4-manifolds

4 流形的几何和拓扑

基本信息

批准号：
16204003
负责人：
MATSUMOTO Yukio
金额：
$ 13.4万
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 (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

Since a decade ago, there have been many research activities on 4-manifolds having the structure of Lefschetz fibrations. The concept of Lefschetz pencils was introduced in algebraic geometry in 1920's, but recently the concept has become the central theme of investigations mainly due to the results of Donaldson and Gompf according to which Lefschetz fibrations are the underlying structure of the symplectic structure. The structure of Lefschetz fibrations is topologically described by their monodromy representations taking the value in the mapping class groups, while the structure is, as mentioned above, closely related to algebraic geometry and symplectic geometry. Thus the study of Lefschetz fibrations is considered to be a meeting place of many research areas, such as 4-manifolds, mapping class groups of surfaces, algebraic geometry, symplectic geometry ad so on. In the present research, we took the above mentioned situation in consideration, and put stress on communication between different research areas. For example, we held a research seminar every summer during the research period entitled "Topology and Algebraic Geometry". Also by the same idea, we held two seminars entitled "Topology and Symplectic Geometry".We have many new results on Lefschetz fibrations, fibering structure of low dimensional manifolds, monodromy, combinatorics of charts, mapping class groups (Teichmuller modular groups), degenerations of Riemann surfaces and the construction of splitting families, topological and analytic local signature, Dedekind sums, and symplectic geometry. By these results, we have obtained some new prospects of research.

十多年来，人们对具有Lefschetz振动结构的4流形进行了大量的研究。Lefschetz铅笔的概念是20世纪20年代在代数几何中引入的，但最近这个概念已经成为研究的中心主题，主要是由于Donaldson和Gompf的结果，根据Lefschetz纤维是辛结构的底层结构。Lefschetz纤维的结构在拓扑上由其在映射类群中取值的单态表示来描述，而如上所述，其结构与代数几何和辛几何密切相关。因此，Lefschetz纤振的研究被认为是许多研究领域的交汇处，如4流形、曲面的映射类群、代数几何、辛几何等。在本研究中，我们考虑到上述情况，并注重不同研究领域之间的交流。例如，在研究期间，我们每年夏天都会举办一个名为“拓扑与代数几何”的研究研讨会。本着同样的思想，我们举办了两次题为“拓扑与辛几何”的研讨会。我们在Lefschetz纤维、低维流形的纤维结构、单一性、图的组合、映射类群（Teichmuller模群）、Riemann曲面的退化和分裂族的构造、拓扑和解析局部签名、Dedekind和、辛几何等方面取得了许多新的成果。通过这些结果，我们得出了一些新的研究前景。