Sheaves on Manifolds pp 217-248 | Cite as

# Micro-support of sheaves

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## Summary

On a manifold *X*, one can naturally associate to any object *F* of **D** ^{ b }(*X*) a closed conic subset of *T*X*, the micro-support of *F*, denoted by SS(*F*). Roughly speaking, SS(*F*) describes the set of codirections of *X* in which *F* “does not propagate”, and we shall prove in the subsequent chapter that SS(*F*) is an involutive subset of *T*X*.

This notion also gives a condition of commutativity of various functors in sheaf theory; e.g. when is *f* ^{!} *F* isomorphic to *ω* _{ Y/X } ⊗ *f* ^{−1} *F* or when is *f* ^{−1} *R* *Hom*(*F* _{1}, *F* _{2}) isomorphic to *RHom*(*f* ^{−1} *F* _{1}, *f* ^{−1} *F* _{2})?

We first prove the equivalence of three definitions of SS(*F*), and use the micro-support to give a criterion in order that, given two open subsets *Ω* _{0} and *Ω* _{1} of *X* with *Ω* _{0} ⊂ *Ω* _{1}, the restriction morphism from *RΓ*(*Ω* _{1}; *F*) to *RΓ*(*Ω* _{0}; *F*) is an isomorphism. The *γ*-topology associated to a closed convex proper cone *γ*, introduced in Chapter III, is a natural tool in the study of the micro-support. In fact if *X* is affine and *φ* _{ γ }: *X* → *X* _{ γ } is the map which weakens the topology of *X*, then *φ* _{ γ } ^{−1} *R* *φ* _{ γ*} plays the role of a cut-off functor in the following sense: SS(*φ* ^{−1} *Rφ* _{ γ*} *F*) is contained in *X* × *γ*°^{ a }, and the morphism *φ* _{ γ } ^{−1} *R* _{ γ*} *F* → *F* is “an isomorphism on *X* × Int *γ*°^{ a }”. This notion of microlocal isomorphism is defined here, and will be developed in the next chapter.

After having given some examples of micro-supports, we study the behavior of micro-supports with respect to various operations on sheaves: tensor product and *Hom*, direct or inverse images, Fourier-Sato transformation.

In this chapter, we always make the assumption that the morphisms are proper or “non-characteristic” with respect to the micro-support. This restriction will be removed in Chapter VI. The results we shall obtain here and in the next chapter, are very similar to many classical results in the theory of partial differential equations with analytic coefficients, and we shall show in Chapter XI, how to deduce some of these classical results from the theory of the micro-support.

We often follow quite tightly the exposition of Kashiwara-Schapira [3], but some proofs are actually simplified.

We keep convention 4.0.

## Keywords

Open Subset Full Subcategory Cotangent Bundle Neighborhood System Natural Morphism## Preview

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