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 sur-face singularities, and from which we use these two invariants to describe when a Gorenstein singularity is quasi-homogeneous.
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