Criteria for Weighted Homogeneity via Logarithmic Vector Fields

Abstract

Recently in [6] the authors proved a lower bound for the $\GSV$ index of a logarithmic vector field along an isolated hypersurface singularity and proposed a conjecture that the weighted homogeneity of this hypersurface singularity can be detected by the existence of either a transverse or a non-degenerate holomorphic logarithmic vector fields. In this paper we prove this conjecture affirmatively. We also prove that the $\GSV$ index attains the lower bound if and only if the hypersurface singularity is weighted homogeneous.

arXiv commentThis new version expands the previous one by further proving that the GSV index attains the minimal value if and only if the isolated hypersurface singularity is weighted homogeneous