Cohomology of regular embeddings
Cohomology of regular embeddings
复制标题
正则嵌入的上同调
DOI：
10.1016/0001-8708(90)90082-x
复制
发表时间：
1990
影响因子：
1.7
通讯作者：
C. Procesi
中科院分区：
文献类型：
--
作者：
Emili Bifet;C. Concini;C. Procesi
本文给出了“完全对称簇”的有理上同环或代数对称簇的正则紧化的显式描述。这种计算的动机来自于描述这些变量的显式舒伯特演算的愿望，自Chasles和Schubert在几个特殊情况下的工作以来，人们一直在考虑这种演算。我们使用的技术是我们试图更好地理解Jurkiewicz和Danilov在环面嵌入方面的工作的结果。用等变上同调的语言对光滑环面嵌入的上同调环的组合表示进行了自然的解释和证明。关键是环面嵌入的Reisner-Stanley代数与它的等变上同调环重合。我们考虑一类空间，正则嵌入，它同时包含“完全对称变体”和光滑环面嵌入。通过适当地推广Reisner-Stanley代数的概念，我们可以显式地计算这些空间的有理等变上同环。
We give in this paper an explicit description of the rational cohomology ring of a “complete symmetric variety” or regular compactification of an algebraic symmetric variety.The motivation for this computation comes from the desire to describe an explicit Schubert calculus for these varieties, which have been considered since the work of Chasles and Schubert in several special cases. The technique we use is the result of our attempts to understand better the work of Jurkiewicz and Danilov in the case of torus embeddings. The combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology. The key point is that the Reisner-Stanley algebra of the torus embedding coincides with its equivariant cohomology ring. We consider a class of spaces, regular embeddings, that contains both “complete symmetric varieties” and smooth torus embeddings. By suitably generalizing the notion of Reisner-Stanley algebra, we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way.