Topological computation of some Stokes phenomena on the affine line

D’Agnolo, Andrea; Hien, Marco; Morando, Giovanni; Sabbah, Claude

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 $\widehat{ℳ}$, 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, $\widehat{ℳ}$ 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 $\widehat{ℳ}$, 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, $\widehat{ℳ}$ 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.

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

[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, Société Mathématique de France (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