The Euler obstruction and the Chern obstruction

Abstract

We determine relations between the Euler obstruction of a map f and the Chern obstruction of a collection of 1-forms associated to f . We obtain explicit applications in singularity theory. On one hand, in [10], Grulha defines a generalization of the Euler obstruction of a holomorphic function to the case of a map-germ f : (V, 0) → (C, 0) where (V, 0) is the germ of a reduced equidimensional complex analytic variety of dimension d ≥ p. The main result in that paper is a formula that computes the Euler obstruction of f in terms of the Euler obstruction for p-frames as defined by Brasselet, Seade and Suwa in [3]. On the other hand, Ebeling and Gusein-Zade introduced in [4] the notion of Chern obstruction for singular spaces using collections of differential 1-forms. This number is well defined for any germ of reduced equidimensional complex analytic space, and the authors provide a formula for the Chern obstruction in terms of intersection number. One of the main results of the present paper is that the Euler obstruction of a map can be written as a Chern obstruction, therefore as an intersection multiplicity.