Towards an Intersection Chow Cohomology Theory for GIT Quotients

TL;DR

This paper develops a new intersection Chow cohomology theory for GIT quotients, showing that operational classes can be represented by strong cycles and establishing a surjective relation from Picard groups to Chow groups.

Contribution

It introduces a novel intersection theory for GIT quotients via topologically strong cycles, extending the understanding of Chow rings for Artin stacks.

Findings

Operational classes are represented by topologically strong cycles.

The map from Picard group to first operational Chow group is surjective with rational coefficients.

The theory applies to good moduli spaces of properly stable, smooth Artin stacks.

Abstract

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are \'etale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, smooth, Artin stack then the natural map from the Picard group of $X$ to the first operational Chow group of $X$ is surjective with rational coefficients.