On non-holonomic irreducible \(D\)-modules
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 \({\mathbb{C}}^ 2\).
scientific article; zbMATH DE number 4075520