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Structure results for torus fixed loci

Jarod Alper, Felix Janda

Abstract

Motivated by localization theorems on moduli spaces, we prove a structural classification of Deligne-Mumford stacks with an action of a torus where the induced action on the coarse moduli space is trivial. We also establish a general local structure theorem for morphisms of algebraic stacks.

Structure results for torus fixed loci

Abstract

Motivated by localization theorems on moduli spaces, we prove a structural classification of Deligne-Mumford stacks with an action of a torus where the induced action on the coarse moduli space is trivial. We also establish a general local structure theorem for morphisms of algebraic stacks.
Paper Structure (21 sections, 12 theorems, 35 equations)

This paper contains 21 sections, 12 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Special cases and our proof strategy
  4. A local structure theorem for morphisms
  5. Examples and Application to gauged linear sigma models
  6. Examples in moduli theory
  7. Stable maps to a Hirzebruch surface
  8. Stable maps to projective space
  9. Relationship to Gauged Linear Sigma Models
  10. Gerbes and torus actions on stacks
  11. Gerbes and rigidification
  12. Group actions on stacks
  13. Fixed locus of a torus action
  14. Universal properties
  15. Extensions of tori and the quotient case
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathcal{X} = [X/\mathbb{G}_m^n] \to B \mathbb{G}_m^n$ be a separated, finite type, and relatively tame Deligne--Mumford morphism of algebraic stacks (i.e., $X$ is a separated, finite type, and tame Deligne--Mumford stack). Assume that $\mathcal{X}$ is reduced and connected, and that the $\math

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2: Torus equivariant interpretation
  • Example 1.3: Case of a classifying stack
  • Example 1.4: Case of a quotient stack
  • Remark 1.5: Root gerbes
  • Remark 1.6
  • Remark 1.7: Strategy of proof in the general case
  • Remark 1.8: Necessity of hypotheses
  • Theorem 1.9
  • Remark 1.10
  • ...and 27 more