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Specialization and microlocalization

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

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].

Keywords

Vector Bundle Open Neighborhood Normal Bundle Horizontal Arrow Normal Deformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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