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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.

##### MSC:
 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
##### Keywords:
ind-sheaves; microsupport; regularity; involutive manifold
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##### References:
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