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 Hhq(V,x) to be the direct limit of Hhq(U-{x}) where U runs over strongly pseudoconvex neighborhoods of x in V, and Hhq(U-{x}) is the holomorphic de Rahm cohomology of the complex manifold U-{x}. 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 p˜x(i) and p¯x(i) of isolated hypersurface singularity (V,x).