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Functorial properties of the microsupport and regularity for ind-sheaves. (English) Zbl 1152.32008
Summary: The notion of microsupport and regularity for ind-sheaves was introduced by M. Kashiwara and P. Schapira in [Astérisque 284, 143–164 (2003; Zbl 1053.35009)]. In this paper we study the behaviour of the microsupport under several functorial operations and characterize “microlocally” the ind-sheaves that are regular along involutive manifolds. As an application, we prove that if a coherent \({\mathcal{D}}\)-module \({\mathcal{M}}\) is regular along an involutive manifold \(V\) [in the sense of M. Kashiwara and T. Oshima, Ann. Math. (2) 106, 145–200 (1977; Zbl 0358.35073)], then the ind-sheaf of temperate holomorphic solutions of \({\mathcal{M}}\) is regular along \(V\). Another application is the notion of microsupport for sheaves on the subanalytic site, which can be given directly, without using the more complicated language of ind-sheaves.

32C38 Sheaves of differential operators and their modules, \(D\)-modules
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32B20 Semi-analytic sets, subanalytic sets, and generalizations
Full Text: DOI arXiv
[1] Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. RIMS Kyoto Univ. 20, 319–365 (1984) · Zbl 0566.32023 · doi:10.2977/prims/1195181610
[2] Kashiwara, M., Oshima, T.: Systems of differential equations with regular singularities and their boundary value problems. Ann. Math. 106, 145–200 (1977) · Zbl 0358.35073 · doi:10.2307/1971163
[3] Kashiwara, M., Schapira, P.: Ind-sheaves. In: Astérisque, vol. 271. Soc. Math. France, Paris (2001) · Zbl 0993.32009
[4] Kashiwara, M., Schapira, P.: Microlocal study of Ind-sheaves I: microsupport and regularity. Astérisque Soc. Math. France 284, 143–164 (2003) · Zbl 1053.35009
[5] Kashiwara, M., Schapira, P.: Sheaves on manifolds, vol. 292. Grundlehren der Math. Wiss. Springer, Heidelberg (1990) · Zbl 0709.18001
[6] Kashiwara, M., Schapira, P.: Categories and sheaves, vol. 332. Springer, Heidelberg (2006) · Zbl 1118.18001
[7] Kashiwara, M., Schapira, P., Ivorra, F., Waschkies, I.: Microlocalization of Ind-sheaves. Birkauser, Boston (2004)
[8] Kashiwara, M., Monteiro Fernandes, T., Schapira, P.: Microsupport and Cauchy problem for temperate solutions of regular \({\mathcal{D}}\) -modules. Portugaliae Mathematica 58(4), 485–504 (2001) · Zbl 1038.35006
[9] Prelli, L.: Sheaves on subanalytic sites. Phd thesis, University Paris, p. 6 (2006). To appear in Rendiconti del Seminario Matematico della Univ. di Padova (arXiv:math.AG/0505498) · Zbl 1171.32002
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