Micro-support and microlocalization

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.