Involutive varieties with smooth support
Let \(X\) be a smooth complex algebraic variety and \(\mathcal D_X\) be its sheaf of rings of differential operators. If \(\mathcal M\) is a coherent sheaf of modules over \(\mathcal D_X\) then its characteristic variety Ch\((\mathcal M)\) is a subvariety of the cotangent bundle \({\text T}^{\star}X.\) A structure theorem for conic involutive varieties of the cotangent bundle of a smooth algebraic variety \(W\) whose projection \(\pi(W)\) on \(X\) is smooth is proved. Let \(W\) be a conical involutive irreducible subvariety of \({\text T}^{\star}X\) and \(\pi(W)\subseteq Y,\) where \(Y\) is a smooth hypersurface of \(X.\) Denote by \(\rho:{\text T}^{\star}X|_Y\to{\text T}^{\star}Y\) the natural projection. Then \(\rho(W)\) is a conical involutive irreducible subvariety of \({\text T}^{\star}Y,\) \(\rho^{-1}(\rho(W))=W\) and \(\dim \rho(W)=\dim W -1.\) The paper includes an example of a conic involutive variety of \(\text{T}^{\star}X\) which is not the characteristic variety of any \(\mathcal D_X\)-module.
sheaves of differential operators and their modules
conic involutive varieties of the cotangent bundle
scientific article; zbMATH DE number 1026087
scientific article