HYPERSURFACE SINGULARITIES WITH 2-DIMENSIONAL CRITICAL LOCUS

Abstract

Consider an analytic germ f:(C^m, 0)→ (C, 0) (m[ges ]3) whose critical locus is a 2-dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f:(C⁴, 0)→(C, 0), defined by f(x₁, x₂, x₃, x₄) = x₁x₂x₃x₄ has the homotopy type of S¹×S¹×S¹. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [9–12, 17, 18].

In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f:(X, x)→(C, 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [8], F has the homotopy type of a bouquet of (dim X−1)-spheres, for example when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F∼F₀∨Sⁿ∨…∨Sⁿ (where F₀ is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F∼∨_iF_i, where the F_i are repeated suspensions of complex links of strata of X. (If (X, x) has the ‘Milnor property’, then the result has been proved by Lê; for details see [6].)

In our situation, the space-germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Sing f⁻¹(0)[les ]2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [12], where dim Sing f⁻¹(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f⁻¹(0) = 2 case. In our investigation some results of Zaharia are crucial [17, 18].