Constructible sheaves

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:

$$\left( {T_{{X_\beta }}^*X\hat + T_{X\alpha }^*X} \right) \cap {\pi ^{ - 1}}\left( {{X_\beta }} \right) \subset T_{{X_\beta }}^*X$$

. (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.