Whitney Equisingularity, Euler Obstruction and Invariants of Map Germs from ℂn to ℂ3, n > 3

Abstract

We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of any finitely determined holomorphic germ f: (ℂn, 0) → (ℂ3, 0), with n > 3. Gaffney showed in [3] that the invariants for the Whitney equisingularity are the 0-stable invariants and the polar multiplicities of the stable types of the germ. First we describe all stable types which appear in these dimensions. Then we find relationships between the polar multiplicities of the stable types in the singular set and also in the discriminant. When n > 3, for any germ f there is an hypersurface in ℂn, which is of special interest, the closure of the inverse image of the discriminant by f, which possibly is with non isolated singularities. For this hypersurface we apply results of Gaffney and Gassler [6], and Gaffney and Massey [7], to show how the Le numbers control the polar invariants of the strata in this hypersurface. Gaffney shows that the number of invariants needed is 4n+10. In the corank one case we reduce this number to 2n+2. The polar multiplicities are also an interesting tool to compute the local Euler obstruction of a singular variety, see [12]. Here we apply this result to obtain explicit algebraic formulae to compute the local Euler obstruction of the stable types which appear in the singular set and also for the stable types which appear in the discriminant, of corank one map germs from ℂn to ℂ3 with n ≥ 3.