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Microlocal study of ind-sheaves. I: Micro-support and regularity. (English) Zbl 1053.35009
Lebeau, Gilles (ed.), On microlocal analysis. Volume dedicated to Jean-Michel Bony on the occasion of his 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-132-5/pbk). Astérisque 284, 143-164 (2003).
In order to treat problems in which one has to handle sheaves associated with, for instance, holomorphic functions with growth conditions, the authors introduced in [Astérisque, 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] the category of ind-sheaves. In the present paper, they give the definition and the elementary properties of the micro-support \(SS(\cdot)\) of ind-sheaves as well as the notion of regularity. In particular, they prove that \(SS(\cdot)\) and the regular micro-support \(SS_{\text{reg}}(\cdot)\) of ind-sheaves behave naturally with respect to distinguished triangles, and that they are invariant through the action of quantized contact transformations. They then apply these results to the ind-sheaves of temperate holomorphic solutions of \({\mathcal D}\)-modules, by proving that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding \({\mathcal D}\)-module and that the ind-sheaf is regular if the \({\mathcal D}\)-module is regular holonomic. They finally compute an example of the ind-sheaf of temperate solutions of an irregular \({\mathcal D}\)-module in dimension one.
For the entire collection see [Zbl 1014.00039].

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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