Intersection cohomology of quotients of nonsingular varieties

TL;DR

This paper presents a method to compute the intersection cohomology of GIT quotients of nonsingular projective varieties by embedding it into equivariant cohomology, facilitating calculations of intersection pairings.

Contribution

It introduces a natural embedding of the intersection cohomology into equivariant cohomology, allowing for explicit computation of intersection pairings in GIT quotients.

Findings

The intersection cohomology embeds into equivariant cohomology under certain conditions.

The intersection pairing can be computed via cup product in equivariant cohomology.

The image of the embedding is characterized by a truncation condition.

Abstract

The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection cohomology as a graded vector space with a nondegenerate pairing.