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arXiv math/9802063math.AG
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等变上同调与等变相交理论

Equivariant cohomology and equivariant intersection theory

Michel Brion

摘要:

本文介绍了等变上同调,拓扑变换群的经典工具,以及等变相交理论,最近由D。Edidin和W.格雷姆它是根据1997年夏天在蒙特利尔举行的讲座编写的。我们的主要目的是获得明确的描述上同调或Chow环的某些流形的群作用出现在表示论,例如齐性空间和它们的紧化。作为等变相交理论的另一个应用,我们得到了舒伯特簇(由于Kumar,Carrell-Peterson和Arabia)的光滑性或有理光滑性准则的简单版本,其陈述和证明在这个框架中变得相当透明。

英文摘要:

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rational smoothness of Schubert varieties (due to Kumar, Carrell-Peterson and Arabia) whose statements and proofs become quite transparent in this framework.

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