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Topology of the Stokes phenomenon

Philip Boalch

TL;DR

The paper develops a unified, intrinsic topological framework for irregular connections by formalizing three equivalent descriptions: Stokes filtrations, Stokes gradings, and Stokes local systems. It proves a simple, canonical splitting theorem: every Stokes filtered local system admits a unique Stokes grading that splits the filtration wherever defined, enabling equivalences among all three viewpoints. This leads to canonical descriptions of wild character varieties and their moduli, via explicit presentations in terms of Stokes representations and graded automorphism actions. The framework elegantly generalizes the Riemann–Hilbert correspondence to irregular singularities and connects to Ramis exponential tori, isomonodromy, and nonabelian Hodge-type structures, with broad implications for wild character varieties and their Poisson/hyperkähler geometry.

Abstract

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy representations. The main result establishes a new simple characterisation of the Stokes decompositions.

Topology of the Stokes phenomenon

TL;DR

The paper develops a unified, intrinsic topological framework for irregular connections by formalizing three equivalent descriptions: Stokes filtrations, Stokes gradings, and Stokes local systems. It proves a simple, canonical splitting theorem: every Stokes filtered local system admits a unique Stokes grading that splits the filtration wherever defined, enabling equivalences among all three viewpoints. This leads to canonical descriptions of wild character varieties and their moduli, via explicit presentations in terms of Stokes representations and graded automorphism actions. The framework elegantly generalizes the Riemann–Hilbert correspondence to irregular singularities and connects to Ramis exponential tori, isomonodromy, and nonabelian Hodge-type structures, with broad implications for wild character varieties and their Poisson/hyperkähler geometry.

Abstract

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy representations. The main result establishes a new simple characterisation of the Stokes decompositions.

Paper Structure

This paper contains 49 sections, 27 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Summary of main result
  3. Further background/other approaches
  4. Generalisations
  5. Layout of the article
  6. Summary of some data canonically determined by a connection
  7. Linear algebra
  8. Gradings
  9. Filtrations
  10. Splittings
  11. Wild monodromy---Relative positions of pairs of gradings
  12. Median gradings
  13. Bounding the wild monodromy
  14. Stokes conditions on pairs of filtrations
  15. Topological basics
  16. ...and 34 more sections

Key Result

Theorem 1.1

For any Stokes filtered local system there is a unique Stokes graded local system (with the same underlying local system) such that the gradings split the filtrations wherever both are defined.

Figures (5)

  • Figure 1: Example Stokes diagram, see §\ref{['ssn: weber']}.
  • Figure :
  • Figure :
  • Figure :
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Theorems & Definitions (43)

  • Theorem 1.1: Splitting
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Example 3.4
  • Lemma 3.5
  • Definition 3.6
  • Remark 3.7
  • Lemma 3.8
  • Definition 3.9
  • ...and 33 more