Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin in Cn. Let be holomorphic and set V=f−1(0). Let Bε be a ball in U of sufficiently small radius ε>0, centred at . We show that f has an associated canonical pencil of real analytic hypersurfaces Xθ, with axis V, which leads to a fibration Φ of the whole space (X∩Bε)∖V over S1. Its restriction to (X∩Sε)∖V is the usual Milnor fibration , while its restriction to the Milnor tube f−1(∂Dη)∩Bε is the Milnor–Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere Sε=∂Bε, 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 obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibre bundle over RP1 with the Xθ 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.