Punctured local holomorphic de Rham cohomology

摘要

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 neighborhoods 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 singularity (V, x) is quasi-homogeneous. We also define and compute various Poincare number p_x~((i)) and p_x~((i)) of isolated hypersurface singularity (V,x).