On Some Topological Properties of Fourier Transforms of Regular Holonomic ${\mathcal{D}}$-Modules
Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 454-468

Voir la notice de l'article provenant de la source Cambridge University Press

We study Fourier transforms of regular holonomic ${\mathcal{D}}$-modules. In particular, we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic ${\mathcal{D}}$-modules will be given. Moreover, we give a new proof of the classical theorem of Brylinski and improve it by showing its converse.
DOI : 10.4153/S0008439519000559
Mots-clés : D-module, Fourier transform, irregular singularity
Ito, Yohei; Takeuchi, Kiyoshi. On Some Topological Properties of Fourier Transforms of Regular Holonomic ${\mathcal{D}}$-Modules. Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 454-468. doi: 10.4153/S0008439519000559
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[BE04] Bloch, S. and Esnault, H., Local Fourier transforms and rigidity for D-modules. Asian J. Math. 8(2004), 587–605. Google Scholar | DOI

[Bry86] Brylinski, J.-L., Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques. Astérisque 140–141(1986), 3–134, 251. Google Scholar

[DHMS17] D’Agnolo, A., Hien, M., Morando, G., and Sabbah, C., Topological computation of some Stokes phenomena on the affine line. Ann. Inst. Fourier, to appear. Google Scholar

[DK16] D’Agnolo, A. and Kashiwara, M., Riemann–Hilbert correspondence for holonomic D-modules. Publ. Math. Inst. Hautes Études Sci. 123(2016), 69–197. https://doi.org/10.1007/s10240-015-0076-y Google Scholar | DOI

[DK17] D’Agnolo, A. and Kashiwara, M., A microlocal approach to the enhanced Fourier–Sato transform in dimension one. Adv. Math. 339(2018), 1–59. https://doi.org/10.1016/j.aim.2018.09.022 Google Scholar | DOI

[Dai00] Daia, L., La transformation de Fourier pour les D-modules. Ann. Inst. Fourier 50(2001), no. 6, 1891–1944. Google Scholar | DOI

[Hei15] Heizinger, H., Stokes structure and direct image of irregular singular D-modules. 2015. Google Scholar

[HR08] Hien, M. and Roucairol, C., Integral representations for solutions of exponential Gauss–Manin systems. Bull. Soc. Math. France 136(2008), 505–532. https://doi.org/10.24033/bsmf.2564 Google Scholar | DOI

[HTT08] Hotta, R., Takeuchi, K., and Tanisaki, T., D-modules, perverse sheaves, and representation theory. Progress in Mathematics, 236, Birkhäuser Boston, Boston, MA, 2008. https://doi.org/10.1007/978-0-8176-4523-6 Google Scholar | DOI

[IT18] Ito, Y. and Takeuchi, K., On irregularities of Fourier transforms of regular holonomic D-Modules. 2018. Google Scholar

[KS90] Kashiwara, M. and Schapira, P., Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, Berlin, 1990. https://doi.org/10.1007/978-3-662-02661-8 Google Scholar | DOI

[KS01] Kashiwara, M. and Schapira, P., Ind-sheaves. Astérisque 271(2001), 1–136. Google Scholar

[KS06] Kashiwara, M. and Schapira, P., Categories and sheaves. Grundlehren der Mathematischen Wissenschaften, 332, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/3-540-27950-4 Google Scholar | DOI

[KS16a] Kashiwara, M. and Schapira, P., Irregular holonomic kernels and Laplace transform. Selecta Math. 22(2016), 55–109. https://doi.org/10.1007/s00029-015-0185-y Google Scholar | DOI

[KS16b] Kashiwara, M. and Schapira, P., Regular and irregular holonomic D-modules. London Mathematical Society Lecture Note Series, 433, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316675625 Google Scholar | DOI

[KL85] Katz, N. S M. and Laumon, G., Transformation de Fourier et majoration de sommes exponentielles. Inst. Hautes Études Sci. Publ. Math. 62(1985), 361–418. Google Scholar | DOI

[Mal88] Malgrange, B., Transformation de Fourier géometrique. Séminaire Bourbaki, Vol. 1987/88. Astérisque 161–162(1989), no. 692, 4, 133–150. Google Scholar

[Moc10] Mochizuki, T., Note on the Stokes structure of Fourier transform. Acta Math. Vietnam 35(2010), 107–158. Google Scholar

[Moc18] Mochizuki, T., Stokes shells and Fourier transforms. 2018. Google Scholar

[Pre11] Prelli, L., Conic sheaves on subanalytic sites and laplace transform. Rend. Semin. Mat. Univ. Padova 125(2011), 173–206. https://doi.org/10.4171/RSMUP/125-11 Google Scholar | DOI

[Rou06] Roucairol, C., Irregularity of an analogue of the Gauss–Manin systems. Bull. Soc. Math. France 134(2006), 269–286. https://doi.org/10.24033/bsmf.2510 Google Scholar | DOI

[Rou07] Roucairol, C., Formal structure of direct image of holonomic D-modules of exponential type. Manuscripta Math. 124(2007), 299–318. https://doi.org/10.24033/bsmf.2510 Google Scholar | DOI

[Sab06] Sabbah, C., Hypergeometric periods for a tame polynomial. Port. Math. 63(2006), 173–226. Google Scholar

[Sab08] Sabbah, C., An explicit stationary phase formula for the local formal Fourier–Laplace transform. In: Singularities I, Contemp. Math., 474, Amer. Math. Soc., Providence, RI, 2008, pp. 309–330.https://doi.org/10.1090/conm/474/09262 Google Scholar | DOI

[Tam08] Tamarkin, D., Microlocal condition for non-displaceability. In: Algebraic and Analytic Microlocal Analysis, AAMA 2013, Springer Proceedings in Mathematics & Statistics, 269, Springer, Cham, 2018, pp. 99–223.https://doi.org/10.1007/978-3-030-01588-6_3 Google Scholar | DOI

[Ver83] Verdier, J.-L., Spécialisation de faisceaux et monodromie modérée. In: Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, 101–102, Soc. Math. France, Paris, 1983, pp. 332–364. Google Scholar

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