Micro-support and microlocalization
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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).
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.
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.
KeywordsNormal Cone Inverse Image Direct Image Cotangent Bundle Natural Morphism
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