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# Micro-support and microlocalization

Chapter
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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 292)

## Summary

Let X be a manifold, Ω a subset of T*X. We define the triangulated category D b (X; Ω) as the localization of D b (X) by the full subcategory of objects whose micro-support is disjoint from Ω. Then to work “microlocally” on Ω with a sheaf F on X gets a precise meaning: it simply means to consider F as an object of D b (X; Ω). With this new notion, we introduce the “microlocal inverse image” and the “microlocal direct image”. These are pro-objects or ind-objects of the category D b (X; p), the localization of D b (X) at p, but we give conditions which ensure that one remains in the category D b (X; p).

The localization of D b (X) is related to the functor μhom by the formula:
$$Ho{m_{{D^b}\left( {X;p} \right)}}\left( {G,F} \right) = {H^0}\left( {\mu \hom \left( {G,F} \right)} \right)$$
(6.0.1)
.

This formula is an essential step in the proof of Theorem 6.5.4 which asserts that SS(F) is an involutive subset of T*X.

Before getting the involutivity theorem, we study the micro-support of sheaves after various operations (direct images for an open embedding, microlocalization, etc.), extending the results of the preceding chapter to the characteristic case, or to the non-proper case. In particular we obtain the formula:
$$SS\left( {\mu \hom \left( {G,F} \right)} \right) \subset C\left( {SS\left( F \right),SS\left( G \right)} \right)$$
(6.0.2)
. This formulation makes use of normal cones in cotangent bundles that we study in §2.

Next we characterize “microlocally” sheaves whose micro-support is contained in an involutive submanifold. In particular, we show that if SS(F) is contained in the conormal bundle to a submanifold Y of X, then F is microlocally isomorphic to the sheaf L γ, for some A-module L.

Finally we investigate the case when the functors of inverse image and that of microlocalization commute, and obtain a sheaf-theoretical version of a result on the Cauchy problem for micro-hyperbolic systems.

## Keywords

Normal Cone Inverse Image Direct Image Cotangent Bundle Natural Morphism
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1990

## Authors and Affiliations

1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyoto 606Japan
2. 2.Department of MathematicsUniversity of Paris VIParis Cedex 05France