Vanishing cycles of \(D\)-modules
type: Item
instance of: scholarly article
title: Vanishing cycles of \(D\)-modules
zbMATH Open document ID: 0799.32031
DOI: 10.1007/BF01232445
author: Yves Laurent
published in: Inventiones Mathematicae
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/144112
review text: Let \(X\) be a complex manifold and \(i:Y \to X\) the inclusion of a smooth hypersurface. If \({\mathcal M}\) is a \({\mathcal D}_ X\) or \({\mathcal E}_ X\)- module, let \(\widetilde \Phi ({\mathcal M}) = i^* (\widetilde {\mathcal D}^ 2 \otimes {\mathcal M})\), where \(\widetilde {\mathcal D}^ 2\) is a sheaf of 2-microdifferential operators and \(i^*\) is the extension to these operators of the inverse image of \({\mathcal D}\)-modules. \(\widetilde\Phi({\mathcal M})\) is a sheaf on the conormal bundle \(\pi:T_ Y^*X\to Y\) and has a \(\pi^{-1} {\mathcal D}_ Y\)-module structure. In fact if \({\mathcal M}\) is specializable (i.e., a \(b\)-function exists for \({\mathcal M})\), then \(\widetilde \Phi ({\mathcal M})\) is coherent as a \(\pi^{- 1} {\mathcal D}_ Y\)-module on \(T_ Y^* X-Y\) and hence locally constant over \(Y\). The equation of \(Y\) gives a nonvanishing section \(\sigma:Y \to T_ Y^* X\) and a coherent \({\mathcal D}_ Y\)-module \(\Phi ({\mathcal M}) = \sigma^{-1} \widetilde \Phi ({\mathcal M})\). This module is proved to be the same as the module of vanishing cycles defined by Kashiwara, Malgrange and Sabbah. This new definition allows one to calculate the characteristic cycle of \(\widetilde \Phi ({\mathcal M})\) from the microcharacteristic cycle of \({\mathcal M}\). In the case when \({\mathcal M}\) is specializable, the author also proves that in the case of an irregular singularity the sheaf \(\widetilde \Phi ({\mathcal M})\) does not change if \(\widetilde {\mathcal D}^ 2\) is replaced by a sheaf \(\widetilde {\mathcal D}^ 2(r)\) of 2-microdifferential operators with specified growth. Then, a Cauchy theorem relates the solutions of the system \({\mathcal M}\) in the sheaf of microfunctions to the holomorphic solutions of \(\widetilde \Phi ({\mathcal M})\). Combining these allows the author to evaluate the growth of solutions of \({\mathcal M}\). The results are extended further to nonsmooth hypersurfaces \(Y\). The proofs make extensive use of ea…
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 58J15
zbMATH DE Number: 549671
zbMATH Keywords: vanishing cycles
zbMATH Keywords: characteristic variety
zbMATH Keywords: Cauchy problem
zbMATH Keywords: microdifferential operators
zbMATH Keywords: \({\mathcal D}\)-modules
MaRDI profile type: Publication
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OpenAlex ID: W426105229
description: scientific article; zbMATH DE number 549671