On the index of a holomorphic vector field tangent to a singular variety

In this article we define and compute an index for a holomorphic vector field on a (possibly singular) subvariety of a complex manifold, provided the subvariety is a local complete intersection. This index reduces to the usual Poincare-Hopf index in case the subvariety is smooth, and is equal more generally to the index defined in [GSV] and [SS]