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Stokes structure of wild difference modules

Yota Shamoto

Abstract

We formulate and prove a Riemann--Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered $\mathscr{A}_{\rm{per}}$-modules. This correspondence is motivated by the Riemann--Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne--Malgrange. It also generalizes the Riemann--Hilbert correspondence for mild difference modules.

Stokes structure of wild difference modules

Abstract

We formulate and prove a Riemann--Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered -modules. This correspondence is motivated by the Riemann--Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne--Malgrange. It also generalizes the Riemann--Hilbert correspondence for mild difference modules.

Paper Structure

This paper contains 40 sections, 22 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Wild difference modules
  3. Wild Stokes structure
  4. Riemann--Hilbert functor
  5. Outline of the paper
  6. Acknowledgments
  7. Notation
  8. Wild Stokes filtration
  9. Definition and fundamental properties
  10. Preliminary
  11. Index sheaf
  12. Wild Stokes-filtered $\mathscr{A}_{{\mathrm{per}}}$-modules
  13. Basic notions and operations
  14. Relation to the mild case
  15. Classification
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\mathsf{Diffc}$ denote the category of difference modules. Let $\mathsf{St}^{\rm wild}(\mathscr{A}_{\mathrm{per}})$ be the category of wild Stokes-filtered $\mathscr{A}_{\mathrm{per}}$-modules. The functor is an equivalence of categories. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (53)

  • Theorem 1.1: Theorem \ref{['main theorem']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 3.1
  • Example 3.2
  • ...and 43 more