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On a smoothness characterization for good moduli spaces

Dan Edidin, Matthew Satriano, Spencer Whitehead

Abstract

Let $\mathcal{X}$ be a smooth Artin stack with properly stable good moduli space $π\colon\mathcal{X} \to X$. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space $X$ is smooth and the morphism $π$ is flat.

On a smoothness characterization for good moduli spaces

Abstract

Let be a smooth Artin stack with properly stable good moduli space . The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space is smooth and the morphism is flat.

Paper Structure

This paper contains 29 sections, 37 theorems, 53 equations, 4 tables.

Table of Contents

  1. Main Results
  2. Introduction
  3. Questions and examples
  4. Outline of the proofs of the main theorems
  5. Theorem \ref{['thm:simple']}: "only if" direction
  6. Theorem \ref{['thm:simple']}: "if" direction
  7. Theorem \ref{['thm:conj-for-tori']}
  8. Representations of simple groups: proof of Theorem \ref{['part:general-facts']}.\ref{['thm:simple']}
  9. Basic facts about pure representations
  10. Proof of the "only if" direction of Theorem \ref{['part:general-facts']}.\ref{['thm:simple']}
  11. Bounding the dimension of a pure representation
  12. Small enough representations
  13. The case of $\mathop{\mathrm{SL}}\nolimits_n$
  14. The case of $\mathop{\mathrm{Sp}}\nolimits_{2n}$
  15. The case of $\mathop{\mathrm{Spin}}\nolimits_{2n+1}$
  16. ...and 14 more sections

Key Result

Theorem 1.3

Let $V$ be an irreducible stable representation of a simple Lie group $G$. Then $V$ is cofree if and only if $V$ is pure.

Theorems & Definitions (88)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.10: Relationship to a result of Brion
  • Remark 1.11: Reducible representations
  • ...and 78 more