EULER OBSTRUCTION, POLAR MULTIPLICITIES AND EQUISINGULARITY OF MAP GERMS IN $\mathcal{O}(n,p), n < p$

Abstract

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs ft:(ℂn,0) → (ℂp,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions. The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.