On the local quotient structure of Artin stacks

TL;DR

This paper demonstrates that Artin stacks with linearly reductive stabilizers are locally quotient stacks near certain points and provides evidence for a broader etale local version of this statement.

Contribution

It establishes local quotient structures for Artin stacks with linearly reductive stabilizers and generalizes known results on stabilizer actions on deformation spaces.

Findings

Artin stacks are locally quotient stacks near points with linearly reductive stabilizers.

The stabilizer acts algebraically on miniversal deformation spaces.

Evidence is provided for the etale local version of the quotient stack conjecture.

Abstract

We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds etale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space generalizing results of Pinkham and Rim.