We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-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.
"A Luna étale slice theorem for algebraic stacks." Ann. of Math. (2) 191 (3) 675 - 738, May 2020. https://doi.org/10.4007/annals.2020.191.3.1