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Good moduli for moduli of objects

Enrico Lampetti

TL;DR

The article constructs good moduli spaces for Toën–Vaquié moduli of objects in finite-type $oldsymbol{ ext{∞}}$-categories with a $t$-structure, and applies the construction to moduli of perverse sheaves. Central to the method are the notions of $ au$-flat objects, the open substack $oldsymbol{ ext{M}}_oldsymbol{ ext C}^{ ext{heartsuit}}$, and explicit criteria for good moduli spaces via $oldsymbol{Θ}$-reductiveness and $oldsymbol{S}$-completeness. The work provides intrinsic, non-proper moduli spaces, including the existence of separated good moduli spaces for closed qc substacks and, in the perverse-sheaf setting, semisimple $oldsymbol{ ext κ}$-points parameterizing simple constituents. In particular, for complex Whitney stratified manifolds and middle perversity, the entire moduli stack of perverse sheaves admits a derived good moduli space, enabling robust geometric and cohomological analyses. The results extend moduli theory beyond proper settings and connect Toën–Vaquié theory with perverse sheaves and stratified-geometry moduli problems, offering new tools for Hodge-theoretic and representation-theoretic applications.

Abstract

We construct good moduli spaces from moduli of objects in the sense of Toën-Vaquié. As an application, we construct good moduli spaces for perverse sheaves.

Good moduli for moduli of objects

TL;DR

The article constructs good moduli spaces for Toën–Vaquié moduli of objects in finite-type -categories with a -structure, and applies the construction to moduli of perverse sheaves. Central to the method are the notions of -flat objects, the open substack , and explicit criteria for good moduli spaces via -reductiveness and -completeness. The work provides intrinsic, non-proper moduli spaces, including the existence of separated good moduli spaces for closed qc substacks and, in the perverse-sheaf setting, semisimple -points parameterizing simple constituents. In particular, for complex Whitney stratified manifolds and middle perversity, the entire moduli stack of perverse sheaves admits a derived good moduli space, enabling robust geometric and cohomological analyses. The results extend moduli theory beyond proper settings and connect Toën–Vaquié theory with perverse sheaves and stratified-geometry moduli problems, offering new tools for Hodge-theoretic and representation-theoretic applications.

Abstract

We construct good moduli spaces from moduli of objects in the sense of Toën-Vaquié. As an application, we construct good moduli spaces for perverse sheaves.

Paper Structure

This paper contains 21 sections, 89 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Perverse sheaves
  3. Constructible sheaves and Stokes data
  4. Linear overview
  5. Notation
  6. Moduli of objects and $t$-structures
  7. Families of objects
  8. Moduli of objects and $t$-structures
  9. An Hartogs-like lemma for flatness
  10. Existence criteria for good moduli spaces
  11. Good moduli spaces for $\mathcal{M}_\mathcal{C}^\heartsuit$
  12. $\Theta$-reductiveness
  13. $\mathrm{S}$-completeness
  14. Existence of good moduli spaces for $\mathcal{M}_\mathcal{C}^\heartsuit$
  15. Quasi-compact substacks of extensions
  16. ...and 6 more sections

Key Result

Theorem 2.0.6

Every closed quasi-compact substack $\mathcal{X} \subset \mathcal{M}_\mathcal{C}^\heartsuit$ admits a separated good moduli space $X$. Moreover, the $\kappa$-points of $X$ parametrize pseudo-perfect semisimple objects of $\mathcal{C}_\kappa^\heartsuit$ lying over $\mathcal{X}$.

Theorems & Definitions (170)

  • Example 2.0.1
  • Example 2.0.3
  • Example 2.0.4
  • Example 2.0.5
  • Theorem 2.0.6: \ref{['good_general']}
  • Theorem 2.0.7: \ref{['good_ext']}
  • Theorem 2.1.1: \ref{['good_Perv_algebraic']}
  • Remark 3.1.2
  • Lemma 3.1.4: AG
  • Definition 3.1.5
  • ...and 160 more