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代数和辛代数的拓扑。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空间被构造为这样的子空间。近年来，辛对偶的拓扑结构利用莫尔斯理论和等变上同调理论得到了广泛的研究.然而，由于超Kahler代数是非紧的，这些方法一般不起作用。所以我们对超Kahler代数的拓扑结构知之甚少，在这个项目的第一年，我们研究了许多由tori提出的超Kahler代数的例子。第二年我们提出了关于它们的上同调的环结构的一个猜想，并给出了部分答案。去年我们肯定地证明了这个猜想。一个超Kahler商包含一个辛对偶的并作为它的变形收缩核。为了证明这个猜想，重要的是要证明这些辛同余以一种简单的方式相交.我们还研究了一些非交换群的超Kahler同余的例子，特别是射影直线上点的位形空间的余切丛的部分紧化.由此我们计算了位形空间上交对的生成函数。

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。