[Microlocalisation des faisceaux sous-analytiques]
On définit la spécialisation et la microlocalisation pour les faisceaux sous-analytiques. En appliquant ces outils aux faisceaux des fonctions holomorphes tempérées et de Whitney, on généralise des constructions classiques. On démontre aussi que les microlocalisations des fonctions holomorphes tempérées et de Whitney ont une structure naturelle de module sur l’anneau des opérateurs microdifférentiels, et sont localement invariants par transformations de contact.
We define the specialization and microlocalization functors for subanalytic sheaves. Applying these tools to the sheaves of tempered and Whitney holomorphic functions, we generalize some classical constructions. We also prove that the microlocalizations of tempered and Whitney holomorphic functions have a natural structure of module over the ring of microdifferential operators, and are locally invariant under contact transformations.
Keywords: Algebraic analysis, specialization, normal deformation, microlocalization, subanalytic sheaves
Mot clés : Analyse algébrique, spécialisation, déformation normale, microlocalisation, faisceaux sous-analytiques
@book{MSMF_2013_2_135__1_0,
author = {Prelli, Luca},
title = {Microlocalization of subanalytic sheaves},
series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
number = {135},
year = {2013},
doi = {10.24033/msmf.445},
mrnumber = {3157166},
zbl = {1295.32018},
language = {en},
url = {http://archive.numdam.org/item/MSMF_2013_2_135__1_0/}
}
Prelli, Luca. Microlocalization of subanalytic sheaves. Mémoires de la Société Mathématique de France, Série 2, no. 135 (2013), 97 p. doi : 10.24033/msmf.445. http://numdam.org/item/MSMF_2013_2_135__1_0/
[1] – Microlocalisation tempérée, Mémoires Soc. Math. France, vol. 57, 1994. | MR | EuDML
[2] – Continuous flows in the plane, Grundlehren der Math., vol. 201, Springer-Verlag, New York-Heidelberg, 1974. | MR | Zbl
[3] & – « Semianalytic and subanalytic sets », Publ. IHÉS 67 (1988), p. 5–42. | MR | EuDML | Zbl | Numdam
[4] – Analytic -modules and applications, Math. Appl., vol. 247, Kluwer Academic Publishers Group, Dordrecht, 1993. | MR
[5] – « Formal microlocalization », C. R. Acad. Sci. Paris Math. 327 (1998), p. 289–293. | MR | Zbl
[6] —, « Specialization of formal cohomology and asymptotic expansions », Publ. RIMS, Kyoto Univ. 37 (2001), p. 37–69. | MR | Zbl
[7] – An introduction to o-minimal geometry, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica.
[8] & – « Leray’s quantization of projective duality », Duke Math. J. 84 (1996), p. 453–496. | MR | Zbl
[9] & – « Sheaves on -topologies », arXiv:1002.0690. | MR
[10] – « DG-methods for microlocalization », Publ. RIMS, Kyoto Univ. 47 (2011), p. 99–140. | MR | Zbl
[11] & – « Multi-specialization and multi-asymptotic expansions », Advances in Math. 232 (2013), p. 432–498. | MR | Zbl
[12] – « The Riemann-Hilbert problem for holonomic systems », Publ. RIMS, Kyoto Univ. 20 (1984), p. 319–365. | MR | Zbl
[13] —, « -modules and microlocal calculus », in Transl. Math. Monogr., Iwanami Series in Modern Math., vol. 217, Amer. Math. Soc., Providence, 2003.
[14] & – Sheaves on manifolds, Grundlehren der Math., vol. 292, Springer-Verlag, Berlin, 1990. | MR
[15] & – Moderate and formal cohomology associated with constructible sheaves, Mémoires Soc. Math. France, vol. 64, 1996. | EuDML | Zbl | Numdam
[16] & – « Ind-sheaves », Astérisque 271 (2001). | Zbl | Numdam
[17] & – « Microlocal study of ind-sheaves I: microsupport and regularity », Astérisque 284 (2003), p. 143–164. | Zbl | Numdam
[18] & – Categories and sheaves, Grundlehren der Math., vol. 332, Springer-Verlag, Berlin, 2006. | Zbl
[19] , , & – « Microlocalization of ind-sheaves », in Studies in Lie theory, Progress in Math., vol. 243, Birkhäuser, 2006, p. 171–221. | MR | Zbl
[20] – « Sur le problème de la division », Studia Mathematica 8 (1959), p. 87–136. | EuDML
[21] – Ideals of differentiable functions, Tata Institute, Oxford University Press, 1967. | MR
[22] —, Équations différentielles à coefficients polynomiaux, Progress in Math., vol. 96, Birkhäuser, 1991.
[23] – « Functorial properties of the microsupport and regularity for ind-sheaves », Math. Zeitschrift 260 (2008), p. 541–556. | MR | Zbl
[24] – « An existence theorem for tempered solutions of -modules on complex curves », Publ. RIMS, Kyoto Univ. 43 (2007), p. 625–659. | MR | Zbl
[25] —, « Tempered solutions of -modules on complex curves and formal invariants », Ann. Inst. Fourier 59 (2009), p. 1611–1639. | MR | EuDML | Zbl | Numdam
[26] – « Sheaves on subanalytic sites », Thèse, Universities of Padova and Paris VI, 2006. | Zbl
[27] —, « Microlocalization of subanalytic sheaves », C. R. Acad. Sci. Paris Math. 345 (2007), p. 127–132. | MR | Zbl
[28] —, « Sheaves on subanalytic sites », Rend. Sem. Mat. Univ. Padova 120 (2008), p. 167–216. | MR | EuDML | Zbl | Numdam
[29] —, « Cauchy-Kowaleskaya-Kashiwara theorem with growth conditions », Math. Zeitschrift 265 (2010), p. 115–124. | MR | Zbl
[30] —, « Microlocalization with growth conditions of holomorphic functions », C. R. Acad. Sci. Paris Math. 348 (2010), p. 1263–1266. | MR | Zbl
[31] —, « Conic sheaves on subanalytic sites and Laplace transform », Rend. Sem. Mat. Univ. Padova 125 (2011), p. 173–206. | MR | EuDML | Zbl | Numdam
[32] – Microdifferential systems in the complex domain, Grundlehren der Math., vol. 269, Springer-Verlag, Berlin, 1985. | MR | Zbl
[33] & – « Index theorem for elliptic pairs », Astérisque 224 (1994).
[34] – « An introduction to -modules », Bull. Soc. Roy. Sci. Liège 63 (1994), p. 223–295. | MR | Zbl
[35] SGA4 – Séminaire Géom. Algébrique du Bois-Marie by M. Artin, A. Grothendieck, J.-L. Verdier, Lecture Notes in Math., vol. 269, Springer-Verlag, Berlin, 1972.
[36] – Linear ordinary differential equation in the complex domain: problems of analytic continuation, Transl. Math. Monogr., vol. 82, Amer. Math. Soc., Providence, 1990. | MR
[37] – Introduction to étale cohomology, Universitext, Springer-Verlag, Berlin, 1994. | MR | Zbl
[38] – Tame topology and o-minimal structures, Lecture Notes Series, vol. 248, Cambridge University Press, Cambridge, 1998. | MR | Zbl
[39] – « Microlocal perverse sheaves », Bull. Soc. Math. de France 132 (2004), p. 397–462. | MR | EuDML | Zbl | Numdam
[40] – « Covering definable open sets by open cells », in O-minimal Structures, M. Edmundo, D. Richardson, and A. Wilkie, eds, Proceedings of the RAAG Summer School Lisbon 2003, Lecture Notes in Real Algebraic and Analytic Geometry, Cuvillier Verlag, 2005.
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