Algebraic rational cells and equivariant intersection theory

Abstract

We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of $\mathbb{Q}$-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any $\mathbb{Q}$-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more generally to spherical varieties. This paper is an extension of arxiv.org/abs/1112.0365 to equivariant Chow groups.Comment: Second version. 24 pages. Substantial changes in the presentation. In particular, the results on Poincar\'e duality (Section 6 of first version) are omitted; they are published in a separate paper (see http://revistas.pucp.edu.pe/index.php/promathematica/article/view/11235