Spectrum of equivariant cohomology as a fixed point scheme

TL;DR

This paper establishes a connection between the equivariant cohomology ring of certain algebraic varieties and the coordinate ring of a fixed point scheme, extending to GKM spaces like toric varieties.

Contribution

It introduces a new perspective linking equivariant cohomology to fixed point schemes for regular actions and generalizes to GKM spaces.

Findings

Equivariant cohomology ring is isomorphic to a fixed point scheme's coordinate ring.

Examples include flag varieties, Schubert varieties, and Bott-Samelson varieties.

Extension to GKM spaces such as toric varieties.

Abstract

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology ring of $X$ is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.