Abstract
Considering a holonomic \({\mathcal{D}}\)-module and a hypersurface, we define a finite family of \({\mathcal{D}}\)-modules on the hypersurface which we call modules of vanishing cycles. The first one had been previously defined and corresponds to formal solutions. The last one corresponds, via Riemann-Hilbert, to the geometric vanishing cycles of Grothendieck-Deligne. For regular holonomic \({\mathcal{D}}\)-modules there is only one sheaf and for non regular modules the sheaves of vanishing cycles control the growth and the index of solutions. Our results extend to non holonomic modules under some hypothesis.
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References
Aoki, T.: Invertibility for microdifferential operators of infinite order, Publ. RIMS Kyoto Univ. 18 (1982), 1–29.
Björk, J.E.: Analytic D-Modules and Applications, Kluwer Acad. Publ., Dordrecht, Boston, London, 1993.
Brylinski, J.L., Dubson, A. and Kashiwara, M.: Formule de l'indice pour lesmodules holonomes et obstruction d'Euler locale, CR Acad. Sci. Paris série I 293 (1981), 573–577.
Brylinski, J.L., Malgrange, B. and Verdier, J.L.: Transformation de Fourier géométrique II, CR Acad. Sci. Paris série I 303 (1986), 193–198.
Bouter de Monvel, L.:Opérateurs pseudodiff érentiels d'ordre infini, Ann. Inst. Fourier Grenoble 22 (1972), 229–268.
Ginsburg, V.: Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327–402.
Grothendieck, A. and Deligne, P.: SGA 7, Vol. 288, Springer, New York, 1973.
Hartshorne, R.: Residues and Duality, Lecture Notes in Math. 20, Springer, New York, 1966.
Honda, N.: On the reconstruction theorem of holonomic modules in the Gevrey classes, Publ. RIMS Kyoto Univ. 27 (1991), 923–943.
Honda, N.: Regularity theorems for holonomic modules, Proc. Japan Acad. Ser. A 69 (1993), 111–114.
Kashiwara, M.: Systems of Microdifferential Equations, Progr.Math. 34, Birkhäuser, Basel, 1983.
Kashiwara, M.: Vanishing Cycles and Holonomic Systems of Differential Equations, Lecture Notes in Math. 1016, Springer, New York, 1983, pp. 134–142.
Kashiwara, M.: Index theorem for constructible sheaves, Astérisque 130 (1985), 193–209.
Kashiwara, M. and Kawaï, T.: Second microlocalization and asymptotic expansions, In: Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys. 126, Springer, New York, 1980, pp. 21–76.
Kashiwara, M. and Kawaï, T.: On the holonomic systems of microdifferential equations III. Systems with regular singularities, Publ. RIMS, Kyoto Univ. 17 (1981), 813–979.
Kashiwara, M. and Schapira, P.: Microlocal study of sheaves, Astérisque 128 (1985).
Kashiwara, M. and Schapira, P.: Sheaves on Manifolds, Grundlehren der Math. 292, Springer, Berlin, 1990.
Laurent, Y.: Théorie de la deuxiéme microlocalisation dans le domaine complexe, Progr. Math. 53, Birkhäuser, Basel, 1985.
Laurent, Y.: Polygone de Newton et b-fonctions pour les modules microdifférentiels, Ann. Ecole Norm. Sup. Ser. 4 20 (1987), 391–441.
Laurent, Y.: Microlocal operators with plurisubharmonic growth, Compositio Math. 86 (1993), 23–67.
Laurent, Y.: Vanishing cycles of D-modules, Invent. Math. 112 (1983), 491–539.
Laurent, Y. and Malgrange, B.: Cycles proches, spécialisation et D-modules, Ann. Inst. Fourier Grenoble 45 (1995).
Malgrange, B.: Polynome de Berstein–Sato et cohomologie évanescente, Analyse et topologie sur les espaces singuliers, Astérisque 101–102 (1983).
Malgrange, B.: Equations différentielles á coefficients polynomiaux, Progr.Math. 96, Birkhäuser, Boston, 1991.
Ramis, J.P.: Dévissage Gevrey, Astérisque 59–60 (1978), 173–204.
Ramis, J.P.: Théorémes d'indices Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc. 48 (1984), 296.
Sabbah, C.: D-modules et cyclesévanescents, géométrie réelle, Travaux en Cours 24, Hermann, Paris, 1987, pp. 53–98.
Sabbah, C. and Mebkhout, Z.: D-Nodules et cyclesévanescents, Travaux en Cours 35, Hermann, Paris, 1988.
Sato, M., Kawaï, T. and Kashiwara, M.: Hyperfunctions and Pseudodifferential Equations, Lect. Notes in Math. 287, Springer, New York, 1980, pp. 265–529.
Sato, M., Kawaï, T. and Kashiwara, M.: Microlocal analysis of theta functions, Adv. Stud. Pure Math. 4 (1984), 267–289.
Schapira, P.: Microdifferential Systems in the Complex Domain, Grundlehren der Math. 269, Springer, Berlin, 1985.
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Laurent, Y. Vanishing Cycles of Irregular D-Modules. Compositio Mathematica 116, 241–310 (1999). https://doi.org/10.1023/A:1000791329695
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DOI: https://doi.org/10.1023/A:1000791329695