Stokes shells and Fourier transforms

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.