×

On irregularities of Fourier transforms of regular holonomic \(\mathcal{D}\)-modules. (English) Zbl 1484.32014

Summary: We study Fourier transforms of regular holonomic \(\mathcal{D}\)-modules. By using the theory of Fourier-Sato transforms of enhanced ind-sheaves developed by Kashiwara-Schapira and D’Agnolo-Kashiwara, a formula for their enhanced solution complexes will be obtained. Moreover we show that some parts of their characteristic cycles and irregularities are expressed by the geometries of the original \(\mathcal{D}\)-modules.

MSC:

32C38 Sheaves of differential operators and their modules, \(D\)-modules
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke Math. J., 73, 2, 269-290 (1994) · Zbl 0804.33013
[2] Ando, K.; Esterov, A.; Takeuchi, K., Monodromies at infinity of confluent A-hypergeometric functions, Adv. Math., 272, 1-19 (2015) · Zbl 1328.14083
[3] Björk, J.-E., Analytic D-Modules and Applications, Mathematics and Its Applications, vol. 247 (1993), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0805.32001
[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] Brylinski, J.-L., Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque, 140-141, 3-134 (1986), 251, Géométrie et analyse microlocales · Zbl 0624.32009
[6] D’Agnolo, A., On the Laplace transform for tempered holomorphic functions, Int. Math. Res. Not. IMRN, 16, 4587-4623 (2014) · Zbl 1304.32006
[7] D’Agnolo, A.; Hien, M.; Morando, G.; Sabbah, C., Topological computation of some Stokes phenomena on the affine line, preprint · Zbl 1505.14019
[8] D’Agnolo, A.; Kashiwara, M., Riemann-Hilbert correspondence for holonomic D-modules, Publ. Math. Inst. Hautes Études Sci., 123, 69-197 (2016) · Zbl 1351.32017
[9] D’Agnolo, A.; Kashiwara, M., A microlocal approach to the enhanced Fourier-Sato transform in dimension one, Adv. Math., 339, 1-59 (2018) · Zbl 1410.32007
[10] Dimca, A., Sheaves in Topology, Universitext (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1043.14003
[11] Ernström, L., Topological Radon transforms and the local Euler obstruction, Duke Math. J., 76, 1, 1-21 (1994) · Zbl 0831.32016
[12] Esterov, A.; Takeuchi, K., Confluent A-hypergeometric functions and rapid decay homology cycles, Amer. J. Math., 137, 2, 365-409 (2015) · Zbl 1321.33018
[13] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Hypergeometric functions and toric varieties, Funkc. Anal. Prilozh., 23, 2, 12-26 (1989)
[14] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math., 84, 2, 255-271 (1990) · Zbl 0741.33011
[15] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants (1994), Birkhäuser: Birkhäuser Boston · Zbl 0827.14036
[16] Goresky, M.; MacPherson, R., Stratified Morse Theory, vol. 3, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0639.14012
[17] Hotta, R.; Takeuchi, K.; Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236 (2008), Birkhäuser: Birkhäuser Boston · Zbl 1136.14009
[18] Ito, Y.; Takeuchi, K., On some topological properties of Fourier transforms of regular holonomic D-modules, Can. Math. Bull. (2020), in press · Zbl 1439.32025
[19] Kashiwara, M., Systems of Microdifferential Equations, Progress in Mathematics, vol. 34 (1983), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, Based on lecture notes by Teresa Monteiro Fernandes translated from the French, with an introduction by Jean-Luc Brylinski · Zbl 0566.32022
[20] Kashiwara, M., Riemann-Hilbert correspondence for irregular holonomic D-modules, Jpn. J. Math., 11, 1, 113-149 (2016) · Zbl 1351.32001
[21] Kashiwara, M.; Schapira, P., Microlocal study of sheaves, Astérisque, 128, 235 (1985) · Zbl 0589.32019
[22] Kashiwara, M.; Schapira, P., Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292 (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0709.18001
[23] Kashiwara, M.; Schapira, P., Integral transforms with exponential kernels and Laplace transform, J. Am. Math. Soc., 10, 4, 939-972 (1997) · Zbl 0888.32004
[24] Kashiwara, M.; Schapira, P., Ind-sheaves, Astérisque, 271, 136 (2001) · Zbl 0993.32009
[25] Kashiwara, M.; Schapira, P., Microlocal study of ind-sheaves. I. Micro-support and regularity, Astérisque, 284, 143-164 (2003) · Zbl 1053.35009
[26] Kashiwara, M.; Schapira, P., Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften, vol. 332 (2006), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1118.18001
[27] Kashiwara, M.; Schapira, P., Irregular holonomic kernels and Laplace transform, Selecta Math., 22, 1, 55-109 (2016) · Zbl 1337.32020
[28] Kashiwara, M.; Schapira, P., Regular and Irregular Holonomic D-Modules, London Mathematical Society Lecture Note Series, vol. 433 (2016), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1354.32008
[29] Katz, N. M.; Laumon, G., Transformation de Fourier et majoration de sommes exponentielles, Inst. Hautes Études Sci. Publ. Math., 62, 361-418 (1985) · Zbl 0603.14015
[30] Kedlaya, K. S., Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., 154, 2, 343-418 (2010) · Zbl 1204.14010
[31] Kedlaya, K. S., Good formal structures for flat meromorphic connections, II: excellent schemes, J. Am. Math. Soc., 24, 1, 183-229 (2011) · Zbl 1282.14037
[32] Kirwan, F., Complex Algebraic Curves, London Mathematical Society Student Texts, vol. 23 (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0744.14018
[33] Malgrange, B., Équations différentielles à coefficients polynomiaux, Progress in Mathematics, vol. 96 (1991), Birkhäuser: Birkhäuser Boston, MA · Zbl 0764.32001
[34] Matsui, Y.; Takeuchi, K., Microlocal study of topological Radon transforms and real projective duality, Adv. Math., 212, 1, 191-224 (2007) · Zbl 1117.14057
[35] Matsui, Y.; Takeuchi, K., Monodromy at infinity of polynomial maps and Newton polyhedra, Int. Math. Res. Not. IMRN, 8, 1691-1746 (2013), (with an appendix by C. Sabbah) · Zbl 1314.32044
[36] Mochizuki, T., Good formal structure for meromorphic flat connections on smooth projective surfaces, (Algebraic Analysis and Around. Algebraic Analysis and Around, Adv. Stud. Pure Math., vol. 54 (2009)), 223-253 · Zbl 1183.14027
[37] Mochizuki, T., Note on the Stokes structure of Fourier transform, Acta Math. Vietnam., 35, 1, 107-158 (2010) · Zbl 1201.32016
[38] Mochizuki, T., Wild harmonic bundles and wild pure twistor D-modules, Astérisque, 340 (2011) · Zbl 1245.32001
[39] Prelli, L., Conic sheaves on subanalytic sites and Laplace transform, Rend. Semin. Mat. Univ. Padova, 125, 173-206 (2011) · Zbl 1239.32009
[40] Sabbah, C., Introduction to algebraic theory of linear systems of differential equations, (D-modules cohérents et holonomes, Élements de la théorie des systèmes différentiels. D-modules cohérents et holonomes, Élements de la théorie des systèmes différentiels, Travaux en Cours, vol. 45 (1993), Hermann: Hermann Paris), 1-80 · Zbl 0841.14014
[41] Sabbah, C., An explicit stationary phase formula for the local formal Fourier-Laplace transform, (Singularities I. Singularities I, Contemp. Math., vol. 474 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 309-330 · Zbl 1162.32018
[42] Saito, M., Irreducible quotients of A-hypergeometric systems, Compos. Math., 147, 2, 613-632 (2011) · Zbl 1219.33012
[43] Schulze, M.; Walther, U., Hypergeometric D-modules and twisted Gauss-Manin systems, J. Algebra, 322, 9, 3392-3409 (2009) · Zbl 1181.13023
[44] Schulze, M.; Walther, U., Resonance equals reducibility for A-hypergeometric systems, Algebra Number Theory, 6, 3, 527-537 (2012) · Zbl 1251.13023
[45] Tamarkin, D., Microlocal condition for non-displaceability, (Algebraic and Analytic Microlocal Analysis. Algebraic and Analytic Microlocal Analysis, Springer Proc. in Math. and Stat., vol. 269 (2018)), 99-223 · Zbl 1416.35019
[46] Verdier, J.-L., Spécialisation de faisceaux et monodromie modérée, (Analysis and Topology on Singular Spaces, II, III. Analysis and Topology on Singular Spaces, II, III, Luminy, 1981. Analysis and Topology on Singular Spaces, II, III. Analysis and Topology on Singular Spaces, II, III, Luminy, 1981, Astérisque, vol. 101 (1983), Soc. Math. France: Soc. Math. France Paris), 332-364 · Zbl 0532.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.