Cohomology of regular embeddings

Abstract

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.