Geometry of moduli spaces and non-abelian localization formal

我们一直在研究超kahler商和辛商的拓扑结构。研究的目的是了解各种重要模空间的结构，因为许多这些空间都被构造成这样的商。近年来，人们利用莫尔斯理论和等变上同调理论对辛商的拓扑进行了深入的研究。然而，由于超kahler商是非紧致的，这些方法通常不能工作。我们对超kahler商的拓扑结构所知甚少。在这个项目的第一年，我们通过tori研究了许多hyperKahler商的例子。第二年，我们提出了关于它们上同调的环结构的猜想，并给出了部分答案。去年我们肯定地证明了这个猜想。超kahler商在其变形收缩时包含一个辛商的并集。为了证明这个猜想，重要的是要证明这些辛商以一种简单的方式相交。我们还研究了非阿贝尔群的超kahler商的一些例子，特别是射影线上点的位形空间的余切束的部分紧化。计算了构型空间上交点对的生成函数。

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.