Geometry of moduli spaces and non-abelian localization formal

模空间的几何和非阿贝尔局部化形式

• 批准号: 09640124
• 负责人: KONNO Hiroshi
• 金额: $ 1.86万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640124/
• 关键词: HyperKahler manifold; toric manifold; symplectic geometry; moment map; モジュライ空間; 安定ベクトル束

We have been studying the topology of hyperKahler quotients and symplectic quotients. The aim of the research is to understand the structures of various important moduli spaces, because many of these spaces are constructed as such kind of quotients. Recently the topology of symplectic quotients has been studied intensively by using Morse theory and equivariant cohomology theory. However, since hyperKahler quotients are non-compact, these methods do not work in general. So little is known about the topology of hyperKahler quotients.In the first year of this project we investigated many examples of hyperKahler quotients by tori. In the second year we proposed a conjecture about the ring structure of their cohomology and gave a partial answer. In the last year we proved the conjecture affirmatively. A hyperKahler quotient contains a union of symplectic quotients as its deformation retract. To prove the conjecture, it is important to show that these symplectic quotients intersect in a simple way.We also investigated some examples of hyperKahler quotients by non-abelian groups, especially a partial compactification of the cotangent bundle of the configuration space of points in the projective line. As a result we calculated the generating function of the intersection parings on the configuration spaces.

我们一直在研究超Kahler商和辛商的拓扑。这项研究的目的是了解各种重要的模空间的结构，因为这些空间中的许多都是作为这类商来构造的。近年来，利用Morse理论和等变上同调理论对辛商的拓扑学进行了深入的研究。然而，由于超Kahler商是非紧的，这些方法一般不起作用。关于超Kahler商的拓扑学，人们知之甚少。在这个项目的第一年，我们用Tori研究了许多超Kahler商的例子。第二年，我们提出了一个关于上同调的环结构的猜想，并给出了部分答案。在去年，我们肯定地证明了这个猜想。当超Kahler商形变收缩时，它包含辛商的并。为了证明这一猜想，证明这些辛商以一种简单的方式相交是很重要的。我们还研究了一些非交换群的超Kahler商的例子，特别是射影直线上的点的配置空间的余切丛的部分紧化。作为结果，我们计算了位形空间上相交配对的母函数。

项目成果

H. Konno: "Cohomology rings of toric hyperkahler manifolds"International Journal of Mathematics. (to appear)(accepted).

H. Konno：“环面超卡勒流形的上同调环”国际数学杂志。

Hiroshi Konno: "Cohomology rings of toric hyperKahler manifolds"International Journal of Mathematics. (accepted).

Hiroshi Konno：“环面超卡勒流形的上同调环”国际数学杂志。