Algebraic rational cells and equivariant intersection theory

TL;DR

This paper introduces algebraic rational cells to advance intersection theory on singular varieties with torus actions, demonstrating that their equivariant Chow groups are freely generated by cell closures, with applications to group embeddings and spherical varieties.

Contribution

It develops the concept of algebraic rational cells and applies it to prove that equivariant Chow groups of certain varieties are freely generated by cell closures, extending previous work to equivariant settings.

Findings

Equivariant Chow groups are freely generated by cell closures in $ ext{Q}$-filtrable varieties.

The notion of algebraic rational cells aids in understanding intersection theory on singular varieties.

Applications include group embeddings and spherical varieties.

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.