[Paper Review] Stokes shells and Fourier transforms

Takuro Mochizuki|arXiv (Cornell University)|Aug 2, 2018

Advanced Topics in Algebra3 citations

TL;DR

This paper introduces Stokes shells as a novel algebraic framework to analyze the Fourier transform of meromorphic flat bundles on the complex line, particularly focusing on the induced transformation of Stokes structures at infinity. It establishes explicit local Fourier transforms for basic building blocks and uses extensions to generalize the results, providing a complete topological and algebraic description of the Fourier transform's action on holonomic D-modules via Stokes data.

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.

Motivation & Objective

• To understand the Fourier transform of algebraic holonomic D-modules through their associated topological data, especially Stokes structures.
• To develop an algebraic framework for computing the local Fourier transform of meromorphic flat bundles with irregular singularities.
• To describe the Stokes structure at infinity of the Fourier transform using explicit algebraic operations.
• To generalize results from basic meromorphic flat bundles to the general case via the notion of extensions of local systems with Stokes structure.
• To establish a complete correspondence between the original D-module data and its Fourier transform using Stokes shells and local Fourier transforms.

Proposed method

• Introduces the concept of Stokes shells as a tool to encode and compute the local Fourier transform of Stokes structures.
• Defines local Fourier transforms for meromorphic flat bundles with irregular singularities at 0 and ∞ using rapid decay and moderate growth homology groups.
• Applies the Riemann-Hilbert correspondence to relate D-modules to constructible sheaves and their cohomological invariants.
• Uses homology and cohomology pairings between Rham classes and homology classes to extract numerical invariants of the Fourier transform.
• Employs extension theory for local systems with Stokes structure to generalize results beyond basic building blocks.
• Establishes duality and compatibility of morphisms across various sheaf-theoretic constructions to ensure consistency of the transform.

Experimental results

Research questions

• RQ1 How does the Fourier transform act on the Stokes structure at infinity of a holonomic D-module?
• RQ2 What algebraic operations describe the local Fourier transform of meromorphic flat bundles with irregular singularities?
• RQ3 How can Stokes shells be used to encode and compute the transformed Stokes structure?
• RQ4 What is the role of extensions in generalizing the local Fourier transform beyond basic building blocks?
• RQ5 How do the homology groups of meromorphic flat bundles relate to the Fourier transform of D-modules?

Key findings

• The local Fourier transform of a meromorphic flat bundle with an irregular singularity at 0 induces a well-defined transformation of the Stokes structure at infinity, which is captured by the introduced Stokes shells.
• Explicit formulas for the local Fourier transform are derived using rapid decay and moderate growth homology groups, providing a computational framework.
• The Fourier transform preserves the duality structure, as shown by the equality of certain morphisms in the derived category, confirming consistency with known duality theorems.
• Extensions of local systems with Stokes structure allow the generalization of the local Fourier transform to arbitrary holonomic D-modules beyond the basic cases.
• The construction leads to a complete topological and algebraic description of the Fourier transform of holonomic D-modules via Stokes data.
• The paper establishes that the Fourier transform of a D-module can be fully reconstructed from its transformed Stokes structure at infinity and the nearby/vanishing cycle data, using the introduced algebraic operations.