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A microlocal approach to the enhanced Fourier-Sato transform in dimension one. (English) Zbl 1410.32007
Let $$V$$ be a one-dimensional complex vector space and $$V^*$$ its dual. The Fourier transform at the Weyl algebra level gives the isomorphism $$z\to-\partial_w$$, $$\partial_z\to w$$ which induces an equivalence between holonomic algebraic $${\mathcal D}$$-modules $${\mathcal M}$$ on $$V$$ and on $$V^*$$ denoted by $${\mathcal M}\to {}^L{\mathcal M}$$. At a microlocal level the Fourier transform is attached to the standard symplectic transform $$\chi:T^* V\to T^* V^*$$. This way a Legendre transform from Puiseux germs on $$V$$ to Puiseux germs on $$V^*$$ appears. The exponential factors of a holonomic algebraic $${\mathcal D}_V$$ module $${\mathcal M}$$ are Puiseux germs on $$V$$ which describe the growth of holomorphic solutions at irregular points. According to the stationary phase formula formulated in Theorem 1.1. the exponential factors of $$^L{\mathcal M}$$ are obtained via Legendre transform from the exponential factors of $${\mathcal M}$$.
The above-mentioned formula can be restated in terms of another realization of the Fourier transform, namely the Fourier-Sato transform for enhanced ind-sheaves. To do this the Riemann-Hilbert correspondence should be used. A microlocal proof of the stationary phase formula is given in the main result of the paper under consideration (Theorem 7.5.1.).

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 30E15 Asymptotic representations in the complex plane
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##### References:
 [1] Abbes, A.; Saito, T., Local Fourier transform and epsilon factors, Compos. Math., 146, 6, 1507-1551, (2010) · Zbl 1206.14034 [2] Abe, T.; Marmora, A., Product formula for p-adic epsilon factors, J. Inst. Math. Jussieu, 14, 2, 275-377, (2015) · Zbl 1319.14025 [3] Arinkin, D., Rigid irregular connections on $$\mathbb{P}^1$$, Compos. Math., 146, 5, 1323-1338, (2010) · Zbl 1200.14035 [4] Bloch, S.; Esnault, H., Local Fourier transforms and rigidity for D-modules, Asian J. Math., 8, 4, 587-605, (2004) · Zbl 1082.14506 [5] D’Agnolo, A., On the Laplace transform for tempered holomorphic functions, Int. Math. Res. Not., 2014, 16, 4587-4623, (2014) · Zbl 1304.32006 [6] D’Agnolo, A.; Hien, M.; Morando, G.; Sabbah, C., Topological computations of some Stokes phenomena, (2017), 51 pp [7] D’Agnolo, A.; Kashiwara, M., Riemann-Hilbert correspondence for holonomic D-modules, Publ. Math. Inst. Hautes Études Sci., 123, 1, 69-197, (2016) · Zbl 1351.32017 [8] D’Agnolo, A.; Kashiwara, M., Enhanced perversities, J. Reine Angew. Math., (2016), ahead of print, 57 pp [9] A. D’Agnolo, M. Kashiwara, Enhanced specialization with applications to the Fourier-Laplace transform in dimension one, in preparation. [10] Deligne, P., Lettre à malgrange du 19/4/1978, (Singularités irregulières, Correspondance et documents, Documents mathématiques, vol. 5, (2007), Société Mathématique de France Paris), 25-26 [11] Fang, J., The principle of stationary phase for the Fourier transform of D-modules, Pacific J. Math., 254, 1, 117-128, (2011) · Zbl 1241.14010 [12] Fu, L., Calculation of ℓ-adic local Fourier transformations, Manuscripta Math., 133, 3-4, 409-464, (2010) · Zbl 1206.14035 [13] García López, R., Microlocalization and stationary phase, Asian J. Math., 8, 4, 747-768, (2004) · Zbl 1100.32005 [14] Graham-Squire, A., Calculation of local formal Fourier transforms, Ark. Mat., 51, 1, 71-84, (2013) · Zbl 1271.14023 [15] Guillermou, S.; Schapira, P., Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, (Homological Mirror Symmetry and Tropical Geometry, Lecture Notes of the Unione Matematica Italiana, vol. 15, (2014), Springer Berlin), 43-85 · Zbl 1319.32006 [16] Hien, M.; Sabbah, C., The local Laplace transform of an elementary irregular meromorphic connection, Rend. Semin. Mat. Univ. Padova, 134, 133-196, (2015) · Zbl 1330.14012 [17] Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 20, 2, 319-365, (1984) · Zbl 0566.32023 [18] Kashiwara, M., D-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217, (2003), Am. Math. Soc. Providence, xvi+254 pp [19] Kashiwara, M., Riemann-Hilbert correspondence for irregular holonomic $$\mathcal{D}$$-modules, Jpn. J. Math., 11, 1, 113-149, (2016) · Zbl 1351.32001 [20] Kashiwara, M.; Schapira, P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292, (1990), Springer Berlin, x+512 pp [21] Kashiwara, M.; Schapira, P., Ind-sheaves, Astérisque, 271, (2001), 136 pp · Zbl 0993.32009 [22] Kashiwara, M.; Schapira, P., Irregular holonomic kernels and Laplace transform, Selecta Math., 22, 1, 55-109, (2016) · Zbl 1337.32020 [23] Kashiwara, M.; Schapira, P., Regular and irregular holonomic D-modules, London Mathematical Society Lecture Note Series, vol. 433, (2016), Cambridge University Press Cambridge, vi+111 pp · Zbl 1354.32008 [24] Katz, N.; Laumon, G., Transformation de Fourier et majoration de sommes exponentielles, Inst. Hautes Études Sci. Publ. Math., 62, 361-418, (1985) · Zbl 0603.14015 [25] Laumon, G., Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math., 65, 131-210, (1987) · Zbl 0641.14009 [26] Malgrange, B., Équations différentielles à coefficients polynomiaux, Progress in Mathematics, vol. 96, (1991), Birkhäuser, vi+232 pp · Zbl 0764.32001 [27] Mochizuki, T., Note on the Stokes structure of Fourier transform, Acta Math. Vietnam., 35, 1, 107-158, (2010) · Zbl 1201.32016 [28] Mochizuki, T., Curve test for enhanced ind-sheaves and holonomic D-modules, (2016), 87 pp [29] Ramero, L., Hasse-arf filtrations in p-adic analytic geometry, J. Algebraic Geom., 21, 1, 97-182, (2012) · Zbl 1236.14028 [30] Sabbah, C., An explicit stationary phase formula for the local formal Fourier-Laplace transform, (Singularities I, Contemp. Math., vol. 474, (2008), Amer. Math. Soc. Providence, RI), 309-330 · Zbl 1162.32018 [31] Sabbah, C., Introduction to Stokes structures, Lect. Notes in Math., vol. 2060, (2013), Springer, xiv+249 pp · Zbl 1260.34002 [32] Sabbah, C., Differential systems of pure Gaussian type, Izv. Ross. Akad. Nauk Ser. Mat., Izv. Math., 80, 1, 189-220, (2016), translation in · Zbl 1339.14011 [33] Tamarkin, D., Microlocal condition for non-displaceability, (2008), 93 pp
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