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Stokes filtered sheaves and differential-difference modules

Yota Shamoto

TL;DR

This work extends the theory of Stokes filtrations to differential-difference modules in two complex variables by introducing Stokes filtered quasi-local systems on the torus $T=(S^1)^2$. It establishes an abelian, strictly-behaved category for good filtrations and provides a geometric bridge between de Rham and Betti data via a two-variable setup, including a detailed analytic framework with growth conditions and period pairings. The main results show that Betti and De Rham data glue into a global Stokes structure with exponential factors determined by sheets of an associated exponential variety, with the gamma-function example concretely illustrating the construction. These developments open avenues toward applications in mirror symmetry and equivariant quantum cohomology, linking Stokes data to broader geometric representation-theoretic frameworks.

Abstract

We introduce the notion of Stokes filtered quasi-local systems. It is proved that the category of Stokes filtered quasi-local systems is abelian. We also give a geometric way to construct Stokes filtered quasi-local systems, which describe the asymptotic behavior of certain classes of solutions to some differential-difference modules.

Stokes filtered sheaves and differential-difference modules

TL;DR

This work extends the theory of Stokes filtrations to differential-difference modules in two complex variables by introducing Stokes filtered quasi-local systems on the torus . It establishes an abelian, strictly-behaved category for good filtrations and provides a geometric bridge between de Rham and Betti data via a two-variable setup, including a detailed analytic framework with growth conditions and period pairings. The main results show that Betti and De Rham data glue into a global Stokes structure with exponential factors determined by sheets of an associated exponential variety, with the gamma-function example concretely illustrating the construction. These developments open avenues toward applications in mirror symmetry and equivariant quantum cohomology, linking Stokes data to broader geometric representation-theoretic frameworks.

Abstract

We introduce the notion of Stokes filtered quasi-local systems. It is proved that the category of Stokes filtered quasi-local systems is abelian. We also give a geometric way to construct Stokes filtered quasi-local systems, which describe the asymptotic behavior of certain classes of solutions to some differential-difference modules.

Paper Structure

This paper contains 52 sections, 59 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Stokes filtered local systems for differential equations
  3. Stokes filtered quasi-local systems
  4. Geometric construction
  5. De Rham cohomology
  6. Betti homology
  7. Main theorem and an example
  8. Further direction related to mirror symmetry
  9. Notations
  10. Stokes filtered quasi-local systems
  11. A sheaf of ordered abelian groups
  12. Pre-Stokes filtrations
  13. Stokes filtrations
  14. Strictness and Abelianity--Statement
  15. Strictness and Abelianity--Proof
  16. ...and 37 more sections

Key Result

Theorem 1.1

The category of non-ramified Stokes filtered local systems is abelian.

Theorems & Definitions (141)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Deligne Deligne, Malgrange Mal, see also Sabbah
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6: See Theorem \ref{['STRICTNESS']} for a more precise statement
  • Theorem 1.7: Theorem \ref{['main theorem']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 131 more