Topological computation of some Stokes phenomena on the affine line
Annales de l'Institut Fourier, to appear, 70 p.

Let ℳ be a holonomic algebraic 𝒟-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier–Laplace transform ℳ ^, including its Stokes multipliers at infinity, in terms of the quiver of ℳ. Let F be the perverse sheaf of holomorphic solutions to ℳ. By the irregular Riemann–Hilbert correspondence, ℳ ^ is determined by the enhanced Fourier–Sato transform F ⋏ of F. Our aim here is to recover Malgrange’s result in a purely topological way, by computing F ⋏ using Borel–Moore cycles. In this paper, we also consider some irregular ℳ’s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.

Soit ℳ un 𝒟-module holonome algébrique sur la droite affine, à singularités régulières y compris à l’infini. Malgrange a donné une description complète de son transformé de Fourier–Laplace ℳ ^, y compris des multiplicateurs de Stokes à l’infini, en termes du carquois de ℳ. Soit F le faisceau pervers des solutions de ℳ. Par la correspondance de Riemann–Hilbert irrégulière, ℳ ^ est déterminé par le transformé de Fourier–Sato enrichi F ⋏ de F. Notre but est de retrouver le résultat de Malgrange de manière purement topologique, en calculant F ⋏ à l’aide de cycles de Borel–Moore. Nous nous intéressons aussi à d’autres 𝒟-modules holonomes irréguliers ℳ, tels que celui provenant de l’équation d’Airy, où les cycles que nous considérons sont reliés aux chemins de plus grande pente.

Received: 2017-07-24
Revised: 2018-07-11
Accepted: 2018-11-06
Classification: 34M40,  44A10,  32C38
Keywords: Perverse sheaf, enhanced ind-sheaf, Riemann–Hilbert correspondence, holonomic D-module, regular singularity, irregular singularity, Fourier transform, quiver, Stokes matrix, Stokes phenomenon, Airy equation, Borel–Moore homology 
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D’Agnolo, Andrea; Hien, Marco; Morando, Giovanni; Sabbah, Claude. Topological computation of some Stokes phenomena on the affine line. Annales de l'Institut Fourier, to appear, 70 p.

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