Punctured local holomorphic de Rham cohomology

Abstract

Let $V$ be a complex analytic space and $x$ be an isolated singular point of $V$. We define the $q$-th punctured local holomorphic de Rham cohomology $H_h^q$$(V,x)$ to be the direct limit of $H_h^q(U-\{x\left\}\right)$ where $U$ runs over strongly pseudoconvex neighborhoods of $x$ in $V$, and $H_h^q(U-\{x\left\}\right)$ is the holomorphic de Rahm cohomology of the complex manifold $U-\left\{x\right\}$. We prove that punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when the singularity $(V,x)$ is quasi-homogeneous. We also define and compute various Poincaré number ${\tilde{p}}_x^{\left(i\right)}$ and ${\overline{p}}_x^{\left(i\right)}$ of isolated hypersurface singularity $(V,x)$.