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Combinatorial structures on Riemann surfaces and topological properties of the moduli space.

黎曼曲面上的组合结构和模空间的拓扑性质。

基本信息

批准号：
14540068
负责人：
OHBA Kiyoshi
金额：
$ 2.3万
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

Our results are as follows :1.We consider Riemann surfaces with Abelian differential constructed from lightning pairs. A lightning pair is a pieacewise linear loop in the complex plane determined by a certain kind of combinatorial data. We give a method of obtaining from the combinatorial data of a lightning pair the genus of the resulting Riemann surface.2.We give a sufficient condition for the completion of the form which is induced from a pre-Tango structure to have non-closed global differential 1-forms. Moreover, we give a lower bound for the dimension of the locus of the curves which have pre-Tango structures inducing such completions, in the moduli space of curves.3.A Haefliger (6,3)-knot means a smoothly embedded 3-sphere in the 6-sphere. We give a definition of unknotting numbers of Haefliger (6,3)-knots geometrically, and determine the unknotting number of each Haefliger (6,3)-knot.4.Twisting the Killing vector fields of certain kind of Kerr-Ads black holes, we reproduce the compact Sasaki-Einstein manifolds constructed by Gauntlett, Martelli, Sparks and Waldram. We also discuss an implication of the twist in string theory and M-theory.5.We construct explicitly a new infinite series of Einstein metrics on the S^3-bundles over S^2, which containing infinite numbers of inhomogeneous ones.

主要结果如下：1.考虑由闪电对构造的具有阿贝尔微分的Riemann曲面。闪电对是由某种组合数据确定的复平面上的分段线性环。我们给出了一种从闪电对的组合数据中得到Riemann曲面亏格的方法. 2.给出了由pre-Tango结构导出的形式的完备化具有非闭整体微分1-形式的充分条件.此外，在曲线的模空间中，我们给出了具有导致这种完备化的pre-Tango结构的曲线的轨迹维数的一个下界。3. Haefliger（6，3）-纽结是指在6-球面中光滑嵌入3-球面。我们给出了Haefliger（6，3）-纽结的解纽结数的几何定义，并确定了每个Haefliger（6，3）-纽结的解纽结数。4.通过对Kerr-Ads黑洞的Killing矢量场的扭曲，我们再现了Gauntlett，Martelli，Sparks和Waldock构造的紧致Sasaki-Einstein流形。5.我们在S^2上的S^3-丛上构造了一个新的Einstein度量的无穷级数，它包含了无穷多个非齐次的Einstein度量。