Generalized Equivariant Cohomology and Stratifications

TL;DR

This paper develops a systematic method for computing generalized equivariant cohomology of stratified complex varieties with torus actions, demonstrated on the affine Grassmannian.

Contribution

It introduces a framework for explicit computation of equivariant cohomology for stratified varieties, extending to various cohomology theories and applying to the affine Grassmannian.

Findings

Explicit formulas for $E_T^*(X)$ as an $E_T^*( ext{pt})$-module.

Applicable to $H_T^*$, $K_T^*$, and $MU_T^*$ cohomology theories.

Successful computation on the affine Grassmannian.

Abstract

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group.