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}) where U runs over strongly pseudoconvex neigh-borhoods of x in V, and H_h^q(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 sin-gularity (V, x) is quasi-homogeneous. We also define and compute various Poincaré number ˜{p}_x⁽ⁱ⁾ and overline{p}_x⁽ⁱ⁾ of isolated hypersurface singularity (V, x).