On non-holonomic irreducible \(D\)-modules
type: Item
instance of: scholarly article
title: On non-holonomic irreducible \(D\)-modules
zbMATH Open document ID: 0658.32009
DOI: 10.1007/BF01394325
author: Joseph Bernstein
author: Valery A. Lunts
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143626
review text: Let \(D=D({\mathbb{C}}^ n)\) be the Weyl algebra. We will consider the filtration on \(D\) determined by the total degree in \(X,\partial\). The first example of a non-holonomic \(D\)-module was constructed by Stafford (1983). It is a \(D\)-module \(D/I\), where \(I\subset D\) is a maximal left ideal generated by \textit{one} operator \(d\in D\) \((n=2)\). In the present paper the authors prove that, for \(n=2\), ``almost each'' operator \(d\in D\) satisfies the same property. Namely, they obtain the following result: Theorem A. Every operator \(d\in D=D({\mathbb{C}}^ 2)\) of degree \(k\geq 4\) with generic symbol (it lies in the complement of a countable number of hypersurfaces) generates a maximal left ideal. Their proof is based on the symplectic geometry of characteristic varieties. Let \(F\) be a holonomic \(D\)-module. Every irreducible component of Ch(\(F\)) is a minimal involutive homogeneous variety (i.h.-variety, for abbrev.) of dimension \(n\). The key points is the existence of higher-dimensional minimal i.h.-variety. They prove the following: Theorem \(A'\). Let \(n=2\), \(k\geq 4\). Then for a generic homogeneous polynomial on \({\mathbb{C}}^{2n}\) of degree \(k\), the variety of its zeroes is a minimal i.h.-variety, for the symplectic structure associated to gr(\(D\)). Theorem \(A'\) easily implies Theorem A. Some conjectures generalizing to higher-dimension theorems A, \(A'\) are given. They also give some results that generalize directly Stafford's example (first order differential operator), based on the non existence of integral algebraic curves for generic vector fields in \({\mathbb{C}}^ 2\).
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 14C99
zbMATH DE Number: 4075520
zbMATH Keywords: involutive variety
zbMATH Keywords: characteristic variety
zbMATH Keywords: \(D\)-module
reviewed by: Luis Narváez-Macarro
MaRDI profile type: Publication
cites work: Q5563439
cites work: Q4065692
cites work: The Integrability of the Characteristic Variety
cites work: Characteristic varieties and vanishing cycles
cites work: Q4143433
description: scientific article; zbMATH DE number 4075520