On the complex formed by contracting differential forms with a vector field on a hypersurface singularity

Abstract

Let (V, 0) subset of (Cn+1, 0) bean analytic hypersurface with an isolated singularity at 0, and X = (X) over tilde \(V) a tangent vector field to V, where (X) over tilde is a holomorphic vector field in (Cn+1, 0) which has an isolated singularity at 0, The homological index of X at 0 can be defined ([4]) as the Euler characteristic of the complex formed by contracting with X the Kahler differentials on V. In that complex, the homology groups are equidimensional and isomorphic to certain modules defined from the finite dimensional C-algebras associated to the jacobian ideal of the function defining V, and to the coordinates of (X) over tilde ([4]). In this paper, we present an algorithm that provides those isomorphisms in an explicit way, so making it possible to face the problem of extending the homological index to other geometric situations ([3]).