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。

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540068/
• 关键词: Riemann surface; Abelian differential; combinatorial structure; genus; Haefliger knot; 高次元knot; Einstein計量; 結び目解消数; 2次微分; 代数曲線

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.

我们的结果如下： 1.我们考虑由闪电对构造的具有阿贝尔微分的黎曼曲面。闪电对是由某种组合数据确定的复平面中的分段线性环。我们给出了一种从闪电对的组合数据中获得黎曼曲面的亏格的方法。2.我们给出了从前 Tango 结构导出的具有非闭合全局的形式的完成的充分条件微分1-形式。此外，我们给出了曲线轨迹维数的下界，该曲线具有在曲线的模空间中诱导此类完成的前探戈结构。3.Haefliger (6,3)-结意味着平滑嵌入的 3- 6 球体中的球体。我们在几何上给出了Haefliger(6,3)-结的解结数的定义，并确定了每个Haefliger(6,3)-结的解结数。4.扭转某类Kerr-Ads黑洞的Killing向量场，我们重现了 Gauntlett、Martelli、Sparks 和 Waldram 构造的紧凑 Sasaki-Einstein 流形。我们还讨论了弦理论和 M 理论中扭曲的含义。 5.我们在 S^2 上的 S^3-丛上显式构造了一个新的无限级数爱因斯坦度量，其中包含无限多个非齐次度量。

项目成果

大場 清: "Genera of Riemann surfaces constructed from lightning polygons"Surikaisekikenkyusho Kokyuroku. (発表予定).

Kiyoshi Ohba：“由闪电多边形构建的黎曼曲面的属”Surikaisekikenkyusho Kokyuroku（即将呈现）。

The unknotting numbers of knotted 3-spheres in the 6-sphere in the Sense of Haefliger

Haefliger 意义上的 6 球体中打结的 3 球体的解结数

• 发表时间: 2004
• 作者: Kiyoshi Ohba

Genera of Riemann surfaces constructed from lightning polygons

由闪电多边形构造的黎曼曲面属

• 发表时间: 2003
• 作者: Kiyoshi Ohba

The unknotting numbers of knotted 3-spheres in the 6-sphere in the sense of Haefliger

Haefliger 意义上的 6 球中打结的 3 球的解结数

• 发表时间: 2004
• 作者: 大場 清

Pre-Tango structures on curves

曲线上的 Pre-Tango 结构

• 发表时间: 2002
• 作者: 横川 光司