Refinements of Milnor's fibration theorem for complex singularities

Abstract

Let X be an analytic subset of an open neighbourhood U of the origin 0 ̲ in C n . Let f : ( X , 0 ̲ ) → ( C , 0 ) be holomorphic and set V = f − 1 ( 0 ) . Let B ε be a ball in U of sufficiently small radius ε > 0 , centred at 0 ̲ ∈ C n . We show that f has an associated canonical pencil of real analytic hypersurfaces X θ , with axis V, which leads to a fibration Φ of the whole space ( X ∩ B ε ) ∖ V over S 1 . Its restriction to ( X ∩ S ε ) ∖ V is the usual Milnor fibration ϕ = f | f | , while its restriction to the Milnor tube f − 1 ( ∂ D η ) ∩ B ε is the Milnor–Lê fibration of f. Each element of the pencil X θ meets transversally the boundary sphere S ε = ∂ B ε , and the intersection is the union of the link of f and two homeomorphic fibres of ϕ over antipodal points in the circle. Furthermore, the space X ˜ obtained by the real blow up of the ideal ( Re ( f ) , Im ( f ) ) is a fibre bundle over RP 1 with the X θ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.