Micro-support and microlocalization

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


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:
$$Ho{m_{{D^b}\left( {X;p} \right)}}\left( {G,F} \right) = {H^0}\left( {\mu \hom \left( {G,F} \right)} \right)$$

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:
$$SS\left( {\mu \hom \left( {G,F} \right)} \right) \subset C\left( {SS\left( F \right),SS\left( G \right)} \right)$$
. 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.


Normal Cone Inverse Image Direct Image Cotangent Bundle Natural Morphism 
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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

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