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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

##### MSC:
 32Sxx Complex singularities 32C38 Sheaves of differential operators and their modules, $$D$$-modules 58J10 Differential complexes
##### Keywords:
vanishing cycles; holonomic $${\mathcal D}$$-module