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Good moduli space for constructible sheaves and Stokes functors

Enrico Lampetti

TL;DR

The paper develops a robust moduli-theoretic framework for two nontrivial families: representations of compact ∞-categories and Stokes data, including their derived enhancements. Building on Alper’s good-moduli-space theory, it proves Θ-reductivity and S-completeness (via Hartogs-type arguments) for the relevant moduli stacks, ensuring the existence of separated good moduli spaces whose κ-points parametrize semisimple objects. It explicitly applies the theory to Rep_k(I)^ aisebox{0pt}{$ lat$}^ ext{heartsuit} and to Stokes functors, producing derived good moduli spaces and, in the case of constructs, deriving exodromy-based identifications with constructible sheaves. The results yield a principled path to quotient-like moduli spaces in noncommutative and irregular-contexts, with quasi-compact substacks and fixed-rank refinements, enabling geometric and cohomological analyses of these objects.

Abstract

In this paper we give construct good moduli spaces for constructible sheaves and Stokes functors. Derived enhancement of such are also considered.

Good moduli space for constructible sheaves and Stokes functors

TL;DR

The paper develops a robust moduli-theoretic framework for two nontrivial families: representations of compact ∞-categories and Stokes data, including their derived enhancements. Building on Alper’s good-moduli-space theory, it proves Θ-reductivity and S-completeness (via Hartogs-type arguments) for the relevant moduli stacks, ensuring the existence of separated good moduli spaces whose κ-points parametrize semisimple objects. It explicitly applies the theory to Rep_k(I)^ aisebox{0pt}{}^ ext{heartsuit} and to Stokes functors, producing derived good moduli spaces and, in the case of constructs, deriving exodromy-based identifications with constructible sheaves. The results yield a principled path to quotient-like moduli spaces in noncommutative and irregular-contexts, with quasi-compact substacks and fixed-rank refinements, enabling geometric and cohomological analyses of these objects.

Abstract

In this paper we give construct good moduli spaces for constructible sheaves and Stokes functors. Derived enhancement of such are also considered.

Paper Structure

This paper contains 18 sections, 44 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Constructible sheaves
  3. 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. Existence criteria for good moduli spaces
  10. Good moduli spaces for $\mathbf{Rep}_k(\mathcal{I})^\heartsuit$
  11. Application: constructible sheaves
  12. Good moduli spaces for Stokes functors
  13. Stokes functors
  14. Sheaves of unramified irregular values
  15. Sheaves of (ramified) irregular values
  16. ...and 3 more sections

Key Result

Theorem 2.0.1

The algebraic stack $\mathbf{Rep}_k(\mathcal{I})^\heartsuit$ admits a good moduli space $\mathop{\mathrm{Rep}}\nolimits_k(\mathcal{I})$ whose $\kappa$-points parametrize semisimple representations of finite dimensional $\kappa$-vector spaces.

Theorems & Definitions (96)

  • Theorem 2.0.1: \ref{['good_Stokes_Arxiv:good_Rep_theorem']}
  • Theorem 2.2.1
  • Remark 3.1.2
  • Lemma 3.1.4: AG
  • Definition 3.1.5
  • Example 3.1.6
  • Definition 3.1.7
  • Definition 3.1.8
  • Remark 3.1.10
  • Definition 3.1.11
  • ...and 86 more