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Specialization and microlocalization

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Sheaves on Manifolds

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

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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].

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© 1990 Springer-Verlag Berlin Heidelberg

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Kashiwara, M., Schapira, P. (1990). Specialization and microlocalization. In: Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften, vol 292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02661-8_6

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  • DOI: https://doi.org/10.1007/978-3-662-02661-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08082-1

  • Online ISBN: 978-3-662-02661-8

  • eBook Packages: Springer Book Archive

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