A Luna \'etale slice theorem for algebraic stacks

TL;DR

This paper proves that algebraic stacks with affine stabilizers are locally equivalent to quotient stacks near points with reductive stabilizers, using an equivariant algebraization approach, with various applications.

Contribution

It establishes an étale local Luna slice theorem for algebraic stacks, extending classical results to a broader stack context with new algebraization techniques.

Findings

Algebraic stacks are étale-locally quotient stacks near points with reductive stabilizers.

The proof employs an equivariant version of Artin's algebraization theorem.

The results have multiple applications in the theory of algebraic stacks.

Abstract

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is \'etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.