no. 2
Topological computation of some Stokes phenomena on the affine line
[Calcul topologique de certains phénomènes de Stokes sur la droite affine]
D’Agnolo, Andrea1 ; Hien, Marco2 ; Morando, Giovanni3 ; Sabbah, Claude4
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
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 739-808.

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.

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.

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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
Mots-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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Topological computation of some {Stokes} phenomena on the affine line},
     journal = {Annales de l'Institut Fourier},
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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, Tome 70 (2020) no. 2, pp. 739-808. doi : 10.5802/aif.3323. https://aif.centre-mersenne.org/articles/10.5802/aif.3323/

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