Riemann–Hilbert and Laplace on the Affine Line (the Regular Case)

Abstract

The Laplace transform \({}^{\mathrm{F}}\!M\) of a holonomic \(\mathcal{D}\)-module M on the affine line \({\mathbb{A}}^{\!1}\) is also holonomic. If M has only regular singularities (included at infinity), \({}^{\mathrm{F}}\!M\) provides the simplest example of an irregular singularity (at infinity). We will describe the Stokes-filtered local system attached to \({}^{\mathrm{F}}\!M\) at infinity in terms of data of M. More precisely, we define the topological Laplace transform of the perverse sheaf \({{}^{\mathrm{p}}\! DR }^{\mathrm{an}}M\) as a perverse sheaf on \(\widehat{{\mathbb{A}}}^{\!1}\) equipped with a Stokes structure at infinity. We make explicit this topological Laplace transform. As a consequence, if \(k\) is a subfield of \(\mathbb{C}\) and if we have a \(k\)-structure on \({{}^{\mathrm{p}}\! DR }^{\mathrm{an}}M\), we find a natural \(k\)-structure on \({{}^{\mathrm{p}}\! DR {}^{\mathrm{an}}}^{\mathrm{F}}\!M\) which extends to the Stokes filtration at infinity. In other words, the Stokes matrices can be defined over \(k\). We end this chapter by analyzing the behaviour of duality by Laplace and topological Laplace transformation, and the relations between them.