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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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 = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Tome 70 (2020) no. 2, pp. 739-808. doi : 10.5802/aif.3323. https://aif.centre-mersenne.org/articles/10.5802/aif.3323/

[1] Balser, Werner; Jurkat, Wolfgang B.; Lutz, Donald A. Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., Volume 71 (1979), pp. 48-94 | DOI | MR | Zbl

[2] Cruz Morales, John A.; van der Put, Marius Stokes matrices for the quantum differential equations of some Fano varieties, Eur. J. Math., Volume 1 (2015) no. 1, pp. 138-153 | DOI | MR | Zbl

[3] D’Agnolo, Andrea On the Laplace transform for tempered holomorphic functions, Int. Math. Res. Not., Volume 2014 (2014) no. 16, pp. 4587-4623 | DOI | MR | Zbl

[4] D’Agnolo, Andrea; Kashiwara, Masaki Riemann–Hilbert correspondence for holonomic D-modules, Publ. Math., Inst. Hautes Étud. Sci., Volume 123 (2016) no. 1, pp. 69-197 | DOI | MR | Zbl

[5] D’Agnolo, Andrea; Kashiwara, Masaki Enhanced perversities, J. reine angew. Math., Volume 751 (2019), pp. 185-241 | DOI | MR | Zbl

[6] Deligne, Pierre Lettre à B. Malgrange du 19/4/1978, Singularités irrégulières, Correspondance et documents (Documents Mathématiques), Volume 5, Société Mathématique de France, 2007, pp. 25-26 | Zbl

[7] Gelfand, Sergei; MacPherson, Robert; Vilonen, Kari Perverse sheaves and quivers, Duke Math. J., Volume 83 (1996) no. 3, pp. 621-643 | DOI | MR | Zbl

[8] Guillermou, Stéphane Le problème de Riemann–Hilbert dans le cas irrégulier [d’après D’Agnolo, Kashiwara, Mochizuki, Schapira], Séminaire Bourbaki (2016/2017) (Astérisque), Volume 407 (2019), pp. 267-296 | DOI | MR

[9] Guillermou, Stéphane; Schapira, Pierre Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, Homological mirror symmetry and tropical geometry (Lecture Notes of the Unione Matematica Italiana), Volume 15, Springer, 2014, pp. 43-85 | DOI | MR | Zbl

[10] Guzzetti, Davide Stokes matrices and monodromy of the quantum cohomology of projective spaces, Commun. Math. Phys., Volume 207 (1999) no. 2, pp. 341-383 | DOI | MR | Zbl

[11] Hien, Marco Topological computations of some Stokes phenomena (joint work with Andrea D’Agnolo, Giovanni Morando and Claude Sabbah), Oberwolfach Reports, Volume 12 (2015) no. 2, pp. 1201-1254 | DOI

[12] Hien, Marco; Sabbah, Claude The local Laplace transform of an elementary irregular meromorphic connection, Rend. Semin. Mat. Univ. Padova, Volume 134 (2015), pp. 133-196 | DOI | Numdam | MR | Zbl

[13] Kashiwara, Masaki The Riemann–Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 2, pp. 319-365 | DOI | MR | Zbl

[14] Kashiwara, Masaki D-modules and Microlocal Calculus, Translations of Mathematical Monographs, 217, American Mathematical Society, 2003, xvi+254 pages | MR | Zbl

[15] Kashiwara, Masaki Riemann–Hilbert correspondence for irregular holonomic D-modules, Jpn. J. Math., Volume 11 (2016) no. 1, pp. 113-149 | DOI | MR | Zbl

[16] Kashiwara, Masaki; Schapira, Pierre Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer, 1990, x+512 pages | MR | Zbl

[17] Kashiwara, Masaki; Schapira, Pierre Ind-sheaves, Astérisque, 271, Société Mathématique de France, 2001, vi+136 pages | Numdam | Zbl

[18] Kashiwara, Masaki; Schapira, Pierre Irregular holonomic kernels and Laplace transform, Sel. Math., New Ser. (2016) no. 1, pp. 55-109 | DOI | MR | Zbl

[19] Kashiwara, Masaki; Schapira, Pierre Regular and irregular holonomic D-modules, London Mathematical Society Lecture Note Series, 433, Cambridge University Press, 2016, vi+111 pages | MR | Zbl

[20] Kedlaya, Kiran Good formal structures for flat meromorphic connections,II: Excellent schemes, J. Am. Math. Soc., Volume 24 (2011), pp. 183-229 | DOI | MR | Zbl

[21] Malgrange, Bernard Equations différentielles à coefficients polynomiaux, Progress in Mathematics, 96, Birkhäuser, 1991 | Zbl

[22] Mebkhout, Zoghman Une équivalence de catégories, Compos. Math., Volume 51 (1984), pp. 55-68 | Numdam | Zbl

[23] Mochizuki, Takuro Note on the Stokes structure of Fourier transform, Acta Math. Vietnam., Volume 35 (2010), pp. 107-158 | MR | Zbl

[24] Mochizuki, Takuro Wild harmonic bundles and wild pure twistor D-modules, Astérisque, 340, Société Mathématique de France, 2011, x+607 pages | Zbl

[25] Prelli, Luca Sheaves on subanalytic sites, Rend. Semin. Mat. Univ. Padova, Volume 120 (2008), pp. 167-216 | DOI | Numdam | MR | Zbl

[26] Prelli, Luca De Rham theorem for Whitney functions, Math. Z. (2019), 10 pages (to appear) | DOI | MR | Zbl

[27] van der Put, Marius; Singer, Michael Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | MR | Zbl

[28] Sabbah, Claude An explicit stationary phase formula for the local formal Fourier–Laplace transform, Singularities I (Contemporary Mathematics), Volume 474, American Mathematical Society, 2008, pp. 309-330 | DOI | MR | Zbl

[29] Sabbah, Claude Introduction to Stokes structures, Lecture Notes in Mathematics, 2060, Springer, 2013 | MR | Zbl

[30] Sabbah, Claude Differential systems of pure Gaussian type, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 80 (2016) no. 1, pp. 201-234 (Russian) translation in Izv. Math. 80 (2016), p. 189-220. | MR | Zbl

[31] Stokes, George G. On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Camb. Philos. Soc., Volume 10 (1857), pp. 105-128 | Zbl

[32] Tamarkin, Dmitry Microlocal condition for non-displaceability, Algebraic and analytic microlocal analysis (Springer Proceedings in Mathematics & Statistics), Volume 269, Springer, 2018, pp. 99-223 | DOI | MR | Zbl

[33] Ueda, Kazushi Stokes matrices for the quantum cohomologies of Grassmannians, Int. Math. Res. Not., Volume 2005 (2005) no. 34, pp. 2075-2086 | DOI | MR | Zbl

Cité par Sources :