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Stokes shells and Fourier transforms

Takuro Mochizuki

Abstract

Algebraic holonomic $\mathcal{D}$-modules on a complex line are classified by the associated topological data consisting of local systems with Stokes structure and the nearby and vanishing cycles at the singularities. The Fourier transform for algebraic holonomic $\mathcal{D}$-modules is defined by exchanging the roles of the variable and the derivative. It is interesting to study the induced transform for the associated topological data. In particular, we closely study the local system with Stokes structure at infinity of the Fourier transform of a $\mathcal{D}$-module, which also allows us to describe the remaining data. We introduce explicit algebraic operations for local systems with Stokes structure, called the local Fourier transform, to study the case of the $\mathcal{D}$-modules associated with basic meromorphic flat bundles. The properties of the local Fourier transforms are captured in terms of Stokes shells. We also introduce the notion of extensions to study the general case.

Stokes shells and Fourier transforms

Abstract

Algebraic holonomic -modules on a complex line are classified by the associated topological data consisting of local systems with Stokes structure and the nearby and vanishing cycles at the singularities. The Fourier transform for algebraic holonomic -modules is defined by exchanging the roles of the variable and the derivative. It is interesting to study the induced transform for the associated topological data. In particular, we closely study the local system with Stokes structure at infinity of the Fourier transform of a -module, which also allows us to describe the remaining data. We introduce explicit algebraic operations for local systems with Stokes structure, called the local Fourier transform, to study the case of the -modules associated with basic meromorphic flat bundles. The properties of the local Fourier transforms are captured in terms of Stokes shells. We also introduce the notion of extensions to study the general case.

Paper Structure

This paper contains 500 sections, 354 theorems, 1473 equations.

Table of Contents

  1. Introduction
  2. Meromorphic flat bundles and local systems with Stokes structure
  3. Local case
  4. Global case
  5. Main purpose in this paper
  6. Fourier transform of $\mathcal{D}$-modules
  7. Main issue in this monograph
  8. Goal
  9. Previous studies
  10. Outline of the introduction
  11. A brief review of Stokes structures
  12. Index sets and special directions
  13. Stokes structures
  14. Lower level Stokes structures and the associated graded objects
  15. Some induced local systems with Stokes structure
  16. ...and 485 more sections

Key Result

Lemma 1.4.3

Theorems & Definitions (396)

  • Remark 1.2.1
  • Remark 1.2.2
  • Remark 1.3.1
  • Definition 1.4.1
  • Definition 1.4.2
  • Lemma 1.4.3
  • Theorem 1.4.4: Theorem \ref{['thm;24.4.5.110']}
  • Proposition 1.4.5
  • Proposition 1.4.6: Proposition \ref{['prop;24.3.25.60']}
  • Lemma 1.4.7
  • ...and 386 more