The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

Abstract

Abstract. Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ fḡ has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ fḡ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of mapgerms of the form fḡ defined in a complex surface germ, and we prove an A'Campo-type formula for the zeta function of its monodromy.

On the Milnor fibration for $f(\mathbf z)\bar g(\mathbf z)$

“…The multiplicity condition has been considered in the paper Fernandez de Bobadilla and Menegon Neto [9] for the case of plane curves. Note that π does not resolve completely the singularities of V (h) but it resolves singularities of V (f ) and V (g).…”

“…Pichon and Seade have studied such functions, especially for the case n = 2 ( [25, 26, 27]). There are also works by Fernandez de Bobadilla and Menegon Neto [9], Parameswaran and Tibar [23], Araujo dos Santos, Ribeiro and Tibar [5], Araujo dos Santos, Ribeiro and Tibar [6], and Joita and Tibar [12]. Note that the link of H is the union of two smooth links defined by f and g respectively which intersect transversely along real codimension 2 smooth variety.…”

The boundary of the Milnor fibre of certain non-isolated singularities

“…Homotopical properties of Y . In order to understand better the structure of Y we consider its fundamental domain (in coordinates (s, t, z)): 1]. The original coordinates are x = e si and y = e ti .…”

“…For several examples in the literature see e.g. [22] (homogeneous singularities, cylinders of plane curves, f = zf ′ (x, y), f = f ′ (x a y b , z)), [26] (f = g(x, y) + zh(x, y)); or for other classes consult also [12] and [1].…”

Topology of smoothings of non-isolated singularities of complex surfaces

“…In collaboration with Weber, they provided explicit plumbing graphs for several classes of singularities: Hirzebruch surface singularities in [MPW07], and the so-called suspensions (f = g(x, y)+ z n ) in [MPW09]. Fernández de Bobadilla and Menegon Neto, in [FdBMN14], proved it in the context of smoothings of non-isolated and not necessarily reduced singularities whose total space has an isolated singularity, for a function of the form f · g, with f and g holomorphic. But none of these approaches was constructive.…”