Let be a holonomic algebraic -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 be the perverse sheaf of holomorphic solutions to . By the irregular Riemann–Hilbert correspondence, is determined by the enhanced Fourier–Sato transform of . Our aim here is to recover Malgrange’s result in a purely topological way, by computing 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 -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 le faisceau pervers des solutions de . Par la correspondance de Riemann–Hilbert irrégulière, est déterminé par le transformé de Fourier–Sato enrichi de . Notre but est de retrouver le résultat de Malgrange de manière purement topologique, en calculant à l’aide de cycles de Borel–Moore. Nous nous intéressons aussi à d’autres -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.
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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
@article{AIF_2020__70_2_739_0, author = {D{\textquoteright}Agnolo, Andrea and Hien, Marco and Morando, Giovanni and Sabbah, Claude}, title = {Topological computation of some {Stokes} phenomena on the affine line}, journal = {Annales de l'Institut Fourier}, pages = {739--808}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {2}, year = {2020}, doi = {10.5802/aif.3323}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3323/} }
TY - JOUR AU - D’Agnolo, Andrea AU - Hien, Marco AU - Morando, Giovanni AU - Sabbah, Claude TI - Topological computation of some Stokes phenomena on the affine line JO - Annales de l'Institut Fourier PY - 2020 SP - 739 EP - 808 VL - 70 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3323/ DO - 10.5802/aif.3323 LA - en ID - AIF_2020__70_2_739_0 ER -
%0 Journal Article %A D’Agnolo, Andrea %A Hien, Marco %A Morando, Giovanni %A Sabbah, Claude %T Topological computation of some Stokes phenomena on the affine line %J Annales de l'Institut Fourier %D 2020 %P 739-808 %V 70 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3323/ %R 10.5802/aif.3323 %G en %F AIF_2020__70_2_739_0
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. http://www.numdam.org/articles/10.5802/aif.3323/
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