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Vanishing cycles and second microlocalization. (English) Zbl 0686.32008
Algebraic analysis, Pap. Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 1, 381-391 (1989).
[For the entire collection see Zbl 0665.00008.]
The aim of this paper is to present a construction of the vanishing cycles of a holonomic \({\mathcal D}\)-module on a hypersurface: first, given a complex manifold X and a smooth hypersurface Y in it, there is defined a sheaf of rings \(\hat {\mathcal D}^ 2_{\Lambda}\) on the canonical bundle \(T^*_ YX\); then it is proved that if \({\mathcal M}\) is a holonomic \({\mathcal D}_ X\)-module, then the module \(\hat {\mathcal M}^ 2\) obtained from \({\mathcal M}\) by extension of the scalars to \(\hat {\mathcal D}^ 2_{\Lambda}\) induces on Y a coherent \({\mathcal D}_ Y\)-module isomorphic to the vanishing cycles of \({\mathcal M}\).
Reviewer: I.Mihut

32Sxx Complex singularities
32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J10 Differential complexes