Vanishing cycles and \(D\)-modules
type: Item
instance of: scholarly article
title: Vanishing cycles and \(D\)-modules
zbMATH Open document ID: 0689.32007
DOI: 10.1007/BF02386373
author: M. G. M. van Doorn
published in: Arkiv för Matematik
publication date: 1989-01-01T00:00:00Z
review text: Let X be a complex manifold, \(f: X\to {\mathbb{C}}\) be a non-constant holomorphic function on X and \(X_ 0:=f^{-1}(0)\). Analogous versions of the Deligne's nearby cycle functor \(\psi_ f\), vanishing cycle functor \(\phi_ f\) aod of natural projections \[ can: \psi_ f\to \phi_ f,\quad var: \phi_ f\to \psi_ f \] are given in this paper. Using the functors \(\phi\) and \(\psi\) defined by Kashiwara on the category of regular holonomic \({\mathcal D}_ X\)-modules with the property that \(\phi{\mathcal M}\) (resp. \(\psi{\mathcal M})\) agree with \(\phi_ f{\mathcal F}\) (resp. \(\psi_ f{\mathcal F})\), \({\mathcal F}=R\hom_{{\mathcal D}_ X}({\mathcal M},{\mathcal O}_ x),{\mathcal M}\in mod({\mathcal D}_ X)_{hr}\), the author prove that the functor \[ F:{\mathcal M}\to ({\mathcal M}[[f^{-1}]],\phi{\mathcal M}\rightleftarrows ^{c}_{v}\psi{\mathcal M},\psi('p)) \] defines an equivalence of categories. Here \(\pi: {\mathcal M}\to {\mathcal M}[[f^{-1}]]\) denotes the canonical map and \(c({\mathcal M})\), v(\({\mathcal M})\) are the natural morphisms corresponding with can and var. By the Riemann-Hilbert correspondance this corresponds to Verdier's extension theorem of perverse sheaves and can be used for the proof. The author prefers to give a proof based only on the language of \({\mathcal D}\)-modules without the Riemann-Hilbert correspondance.
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 35A27
zbMATH DE Number: 4129186
zbMATH Keywords: vanishing cycle functor
zbMATH Keywords: regular holonomic \({\mathcal D}_ X\)-modules
zbMATH Keywords: Riemann-Hilbert correspondance
reviewed by: Stancho Dimiev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02386373
OpenAlex ID: W2314311052
description: scientific article; zbMATH DE number 4129186