A vanishing theorem for a class of systems with simple characteristics
type: Item
instance of: scholarly article
title: A vanishing theorem for a class of systems with simple characteristics
zbMATH Open document ID: 0626.58028
DOI: 10.1007/BF01388871
author: Masaki Kashiwara
author: Pierre Schapira
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143301
review text: Let \((X,O_ X)\) be a complex manifold of dimension n, and let \(E_ x\) be the sheaf of finite-order microdifferential operators. Let M be a coherent \(E_ X\)-module defined on an open subset U of \(T^*X\), and let \(V=char(M)\) be its characteristic variety. Let M be a real analytic manifold in X, such that X is a complexification of M, and let \(C_ M\) be the sheaf of Sato's microfunctions on the conormal bundle \(T^*_ MX\) to M in X. Assume that \(V\cap T^*_ MX\) is smooth, \(T(V\cap T^*_ MX)=TV\cap TT^*_ MX\), and the canonical 1-form \(\omega\) does not vanish on \(V\cap T^*_ MX\). Let \(L_{T^*_ MX}(V)\) be the Levi form of V with respect to \(T^*_ MX\). Then, the authors prove the following: Theorem. If the number of negative eigenvalues of \(L_{T^*_ MX}(V)\) is constant on \(V\cap T^*_ MX\), say r, then the groups \(Ext^ j_{E_ x}(M,C_ M)\) are zero for \(j\neq r\).
Mathematics Subject Classification ID: 58J99
zbMATH DE Number: 4017964
zbMATH Keywords: finite-order microdifferential operators
zbMATH Keywords: Sato's microfunctions on the conormal bundle
reviewed by: Kazuo Taniguchi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4017964