Local holomorphic De Rham cohomology

Local holomorphic De Rham cohomology introduced in this paper and punctured local holomorphic De Rham cohomology introduced by Huang-Luk-Yau are two important local invariants for varieties with isolated singularities. We find some relations between these two invariants and the invariants defined by Steenbrink on surface singularities, and from which we use these two invariants to describe when a Gorenstein singularity is quasi-homogeneous.