Specialization and microlocalization

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 292)


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].


Vector Bundle Open Neighborhood Normal Bundle Horizontal Arrow Normal Deformation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyoto 606Japan
  2. 2.Department of MathematicsUniversity of Paris VIParis Cedex 05France

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