Specialization and microlocalization

Summary

Let X be a manifold, M a closed submanifold. We first construct a new manifold ${\tilde X_M}$, the normal deformation of M in X. This manifold is of dimension one more than the dimension of X, and is endowed with a map $\left( {p,t} \right):{\tilde X_M} \to X \times \mathbb{R}$ such that t ⁻¹ (c) is isomorphic to X for c ≠ 0 and t ⁻¹(0) is isomorphic to T _M X, the normal bundle to M in X.

We use this manifold to associate to a sheaf F on X (or more generally to F ∈ Ob(D ^b(X))) an object v _M (F) of D ^b(T _M X) called the specialization of F along M. Its Fourier-Sato transform μ _M (F) is the microlocalization of F along M.

Having defined the functors v _M and μ _M and studied their functorial properties, we then proceed to study the functor µhom.

This new functor generalizes the microlocalization functor and will play a central role throughout the rest of the book. The results of §2 and §3 originated from Sato-Kawai-Kashiwara [1].