Brieskorn modules and Gauss-Manin systems for non-isolated hypersurface singularities

Abstract

We study the Brieskorn modules associated to a germ of a holomorphic function with non-isolated singularities and show that the Brieskorn module has naturally the structure of a module over the ring of microdifferential operators of non-positive degree, and that the kernel of the morphism to the Gauss–Manin system coincides with the torsion part for the action of t and also with that for the action of the inverse of the Gauss–Manin connection. This torsion part is not finitely generated in general, and a sufficient condition for the finiteness is given here. A Thom–Sebastiani-type theorem for the sheaf of Brieskorn modules is also proved when one of two functions has an isolated singularity.