Logarithmic comparison theorem versus Gauss-Manin system for isolated singularities

TL;DR

This paper explores the conditions under which the logarithmic comparison theorem applies to isolated hypersurface singularities, linking it to the Gauss-Manin system and monodromy eigenvalues, especially in non-quasihomogeneous cases.

Contribution

It provides a necessary condition involving the Gauss-Manin system for the logarithmic comparison theorem to hold in non-quasihomogeneous singularities.

Findings

Logarithmic comparison theorem holds only if 1 is an eigenvalue of monodromy in non-quasihomogeneous cases.

Explicit characterization of the theorem for quasihomogeneous singularities.

Necessary condition for the theorem in non-quasihomogeneous singularities.

Abstract

For quasihomogeneous isolated hypersurface singularities, the logarithmic comparison theorem has been characterized explicitly by Holland and Mond. In the non quasihomogeneous case, we give a necessary condition for the logarithmic comparison theorem in terms of the Gauss-Manin system of the singularity. It shows in particular that the logarithmic comparison theorem can hold for a non quasihomogeneous singularity only if 1 is an eigenvalue of the monodromy.