Topological computation of some Stokes phenomena on the affine line
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 739-808.

Let 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 ^, including its Stokes multipliers at infinity, in terms of the quiver of . Let F be the perverse sheaf of holomorphic solutions to . By the irregular Riemann–Hilbert correspondence, ^ 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 ’s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.

Soit 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 ^, y compris des multiplicateurs de Stokes à l’infini, en termes du carquois de . Soit F le faisceau pervers des solutions de . Par la correspondance de Riemann–Hilbert irrégulière, ^ 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 , tels que celui provenant de l’équation d’Airy, où les cycles que nous considérons sont reliés aux chemins de plus grande pente.

Published online:
DOI: 10.5802/aif.3323
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
Mot clés : Faisceau pervers, ind-faisceau enrichi, correspondance de Riemann–Hilbert, D-module holonome, singularité régulière, singularité irrégulière, transformation de Fourier, carquois, matrice de Stokes, phénomène de Stokes, équation d’Airy, homologie de Borel–Moore
D’Agnolo, Andrea 1; Hien, Marco 2; Morando, Giovanni 3; Sabbah, Claude 4

1 Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy
2 Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
3 I.T.T. Marconi, Padova, Italy
4 CMLS, École polytechnique, CNRS, Université Paris-Saclay F–91128 Palaiseau cedex France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Topological computation of some {Stokes} phenomena on the affine line},
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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, Volume 70 (2020) no. 2, pp. 739-808. doi : 10.5802/aif.3323.

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