## Summary

We begin this chapter by explaining the notion of constructible sheaves on a simplicial complex, then by recalling some basic facts about Hironaka’s sub-analytic sets.

Next we introduce the following definition, a modification of that of a Whitney stratification. A stratification *X* = ⨆_{
α ∈ A
}
*X*
_{
α
} of a real analytic manifold *X* is a *μ*-stratification if the *X*
_{
α
}’s are subanalytic submanifolds, and for each pair (*α*, *β*) with \(X\beta \cap \bar X\alpha \ne \emptyset \), we have \({X_\beta } \subset {\bar X_\alpha }\) and also:

. (Recall the operation \(\hat + \) of VI §2.) One proves that given a closed conic sub-analytic isotropic subset *Λ* of *T***X*, there exists a *μ*-stratification *X* = ⨆_{
α
}
*X*
_{
α
} such that \(\Lambda \subset { \sqcup _\alpha }T_{{X_\alpha }}^*X\).

Next we introduce the following definitions. An object *F* of **D**
^{b}(*X*) is weakly ℝ-constructible (*w*-ℝ-constructible, for short) if there exists a locally finite covering *X* = ⋃_{
j
}
*X*
_{
j
} by subanalytic subsets such that for all *k* and all *j*, the sheaves \({H^k}\left( F \right){|_X}_j\), are locally constant. If moreover the complexes *F*
_{
x
} are perfect for all *x* ∈ *X*, one says *F* is ℝ-constructible.

Using the existence of *μ*-stratifications, we prove that *F* is *w*-ℝ-constructible if and only if SS(*F*) is contained in a closed conic subanalytic isotropic set, or equivalently if SS(*F*) is subanalytic and Lagrangian. In other words, to be *w*-ℝ-constructible is a microlocal property. Then we can make full use of the results of the preceding chapters to obtain quite immediately various functorial properties of *w*-ℝ and ℝ-constructible objects. Using the existence of triangulations for subanalytic sets, we also prove that the derived category of ℝ-constructible sheaves is equivalent to the full subcategory of **D**
^{b}(*X*) consisting of objects with ℝ-constructible cohomology.

When *X* is a complex manifold, one can also introduce the notions of *w*-ℂ and ℂ-constructible objects. The definitions are similar, replacing “subanalytic submanifold” by “complex analytic submanifold”. A useful result asserts that *F* is *w*-ℂ-constructible if and only if *F* is *w*-ℝ-constructible (on the real underlying manifold) and moreover SS(*F*) is invariant by the action of ℂ^{×} on *T***X*. By this theorem, many properties of ℂ-constructible sheaves are deduced from those of ℝ-constructible sheaves.

Finally we introduce the famous “nearby-cycle functor” and “vanishing-cycle functor” and compare them to the specialization functor and the microlocalization functor.

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© 1990 Springer-Verlag Berlin Heidelberg

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Kashiwara, M., Schapira, P. (1990). Constructible sheaves. In: Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften, vol 292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02661-8_10

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DOI: https://doi.org/10.1007/978-3-662-02661-8_10

Publisher Name: Springer, Berlin, Heidelberg

Print ISBN: 978-3-642-08082-1

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