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The derived moduli of Stokes data

Mauro Porta, Jean-Baptiste Teyssier

TL;DR

This work develops a comprehensive derived-algebraic framework for Stokes data associated to flat bundles on complex varieties. By recasting Stokes structures as functors on exit-path categories of exodromic stratified spaces and assembling them into hyperconstructible hypersheaves, the authors establish that the moduli of Stokes data forms a locally geometric derived stack locally of finite presentation, valid in arbitrary dimension and with ∞-categorical coefficients. Central innovations include the Stokes functor formalism, exodromy, level structures (and their ramified variants), spreading-out principles, and a Toën–Vaquié-type moduli-objects approach, yielding stability, finite-type results, and a geometric moduli space description. The results connect classical Stokes filtrations in dimension one with higher-dimensional, derived, and universal moduli spaces, providing a robust foundation for shifted-symplectic structure expectations and noncommutative geometry interpretations of Stokes data.

Abstract

The goal of this paper is to show that Stokes data coming from flat bundles form a locally geometric derived stack locally of finite presentation. This generalizes existing geometricity results on Stokes data in four different directions: our result applies in any dimension, $\infty$-categorical coefficients are allowed, derived structures on moduli spaces are considered and more general spaces than those arising from flat bundles are permitted.

The derived moduli of Stokes data

TL;DR

This work develops a comprehensive derived-algebraic framework for Stokes data associated to flat bundles on complex varieties. By recasting Stokes structures as functors on exit-path categories of exodromic stratified spaces and assembling them into hyperconstructible hypersheaves, the authors establish that the moduli of Stokes data forms a locally geometric derived stack locally of finite presentation, valid in arbitrary dimension and with ∞-categorical coefficients. Central innovations include the Stokes functor formalism, exodromy, level structures (and their ramified variants), spreading-out principles, and a Toën–Vaquié-type moduli-objects approach, yielding stability, finite-type results, and a geometric moduli space description. The results connect classical Stokes filtrations in dimension one with higher-dimensional, derived, and universal moduli spaces, providing a robust foundation for shifted-symplectic structure expectations and noncommutative geometry interpretations of Stokes data.

Abstract

The goal of this paper is to show that Stokes data coming from flat bundles form a locally geometric derived stack locally of finite presentation. This generalizes existing geometricity results on Stokes data in four different directions: our result applies in any dimension, -categorical coefficients are allowed, derived structures on moduli spaces are considered and more general spaces than those arising from flat bundles are permitted.

Paper Structure

This paper contains 50 sections, 118 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. Splitting condition.
  3. Induction condition.
  4. Stratified spaces and constructible sheaves
  5. Atomic generation
  6. Stratifications and hyperconstructible hypersheaves
  7. Exodromic stratified spaces
  8. Triangulations and hereditary finality
  9. Subanalytic stratified space
  10. Stokes stratified spaces
  11. Stokes stratified spaces
  12. Stokes loci
  13. The filtered and the Stokes hyperconstructible hypersheaves
  14. The hyperconstructible hypersheaves of filtered functors
  15. The hyperconstructible hypersheaves of Stokes functors
  16. ...and 35 more sections

Key Result

Theorem 1.6

Let $k$ be a (possibly animated) commutative ring. In the setting of Moduli_Stokes_arxiv:situation_intro, the derived prestack defined by the rule is locally geometric locally of finite presentation. Moreover, for every animated commutative $k$-algebra $A$ and every morphism classifying a Stokes functor $F \colon \mathcal{I} \to \mathrm{Perf}_A$, there is a canonical equivalence where $\mathbb

Theorems & Definitions (321)

  • Example 1.2
  • Remark 1.4
  • Remark 1.5: Stokes functors vs. Stokes filtered local systems
  • Theorem 1.6: \ref{['Moduli_Stokes_arxiv:geometricit_Stokes_classical_case']}
  • Theorem 1.7: \ref{['Moduli_Stokes_arxiv:stability_lim_colim_ISt']}
  • Theorem 1.8: \ref{['Moduli_Stokes_arxiv:thm:St_presentable_stable']} and \ref{['Moduli_Stokes_arxiv:Grothendieck_abelian']}
  • Theorem 1.9: \ref{['Moduli_Stokes_arxiv:finite_typeness']}
  • Theorem 1.11: \ref{['Moduli_Stokes_arxiv:thm:moduli_Stokes_vector_bundles']}
  • Definition 1.12
  • Definition 1.13
  • ...and 311 more