Characteristic varieties and vanishing cycles
type: Item
instance of: scholarly article
title: Characteristic varieties and vanishing cycles
zbMATH Open document ID: 0598.32013
DOI: 10.1007/BF01388811
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143343
review text: The main theme of this paper is a systematical study on characteristic variety of a holonomic system with regular singularity on a complex manifold X, especially extensions of index theorems on characteristic cycles. A holonomic system of \({\mathcal D}_ X\)-modules is originally introduced by Sato-Kawai-Kashiwara in 1972, which is a \({\mathcal D}_ X\)- module \({\mathcal M}\) whose dimension of its characteristic variety SS(\({\mathcal M})\) coincides with the dimension of X. For two holonomic systems \({\mathcal N}\) and \({\mathcal M}\) with regular singularity, the author of this paper proves a global index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})=I(SS({\mathcal M}),SS({\mathcal N}))\) where I(A,B) is the multiplicity of the intersection of the cycles A and B. Moreover the author proves the local index theorem: \(\Sigma_ k(- 1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})_ X=I_ X(SS({\mathcal M}),SS({\mathcal N})),\) and the microlocal index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal E}_ X^{{\mathbb{R}}}}({\mathcal N}^{{\mathbb{R}}},{\mathcal M}^{{\mathbb{R}}})_{\xi}=I_{\xi}(SS({\mathcal M}),SS({\mathcal N})).\) The present work arise from the author's interest to characteristic varieties of the highest-weight modules over complex semi-simple Lie algebras. The detailed treatment of that subject appears in the next paper [Adv. Math. 61, 1-48 (1986)].
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 3964374
zbMATH Keywords: index theorems on characteristic cycles
zbMATH Keywords: holonomic system of \({\mathcal D}_ X\)-modules
zbMATH Keywords: characteristic varieties
author: Victor Ginzburg
reviewed by: Masakazu Muro
MaRDI profile type: Publication
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OpenAlex ID: W2051256605
description: scientific article; zbMATH DE number 3964374