A Luna étale slice theorem for algebraic stacks

Abstract

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.

47 pages, reorganization of results and applications, corrected applications to Bialynicki-Birula decompositions and equivariant versal deformations for curves, additional material added throughout, final version

Cited by in corpus (47)

• Existence of moduli spaces for algebraic stacks
• Reductivity of the automorphism group of K-polystable Fano varieties
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• The étale local structure of algebraic stacks
• Hall algebras in the derived category and higher rank DT invariants
• Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves
• Virtual cycles of stable (quasi)-maps with fields
• K-theory and G-theory of derived algebraic stacks
• Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
• On toric geometry and K-stability of Fano varieties
• The telescope conjecture for algebraic stacks
• The Abelian-Nonabelian Correspondence for $I$-functions
• Intersections of two Grassmannians in ${\mathbf{P}}^9$
• Canonical reduction of stabilizers for Artin stacks with good moduli spaces
• On K-stability of some del Pezzo surfaces of Fano index 2
• Functorial destackification and weak factorization of orbifolds
• Models of Brauer-Severi surface bundles
• Purity and 2-Calabi-Yau categories
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• Towards an Intersection Chow Cohomology Theory for GIT Quotients
• Enumerative invariants and wall-crossing formulae in abelian categories
• Addendum: Étale dévissage, descent and pushouts of stacks
• Reductive quotients of klt singularities
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• A Theory of Stacky Fans
• K-moduli of Fano 3-folds can have embedded points
• Noncommutative resolutions and intersection cohomology for quotient singularities
• Uniformization of semistable bundles on elliptic curves
• Computing isogenies from modular equations in genus two
• A blowup formula for virtual enumerative invariants on projective surfaces
• Spaces with ${\mathbb{G}}_m$-action, hyperbolic localization and nearby cycles
• Strong cycles and intersection products on good moduli spaces
• A generalization of Luna's fundamental lemma for stacks with good moduli spaces
• A motivic integral identity for $(-1)$-shifted symplectic stacks
• Intrinsic Stabilizer Reduction and Generalized Donaldson-Thomas Invariants
• Artin algebraization and quotient stacks
• Behrend's function is not constant on ${\mathrm{Hilb}}^n({\mathbb{A}}^3)$
• On window theorem for categorical Donaldson-Thomas theories on local surfaces and its applications
• Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories
• Rational singularities for moment maps of totally negative quivers
• The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory
• K-moduli of Fano threefolds in family 3.10
• Bounding the minimal number of generators of an Azumaya algebra
• Good Hilbert functors
• Cohomological integrality for weakly symmetric representations of reductive groups
• The integral identity conjecture in motivic homotopy theory
• The $\mathrm{Br}={\mathrm{Br}}^{\prime }$ question for some classifying stacks