Refinements of Milnor's fibration theorem for complexsingularities

摘要

Let X be an analytic subset of an open neighbourhood U of the origin 0 in C~n.Letf : (X,0)→(C, 0)be holomorphic and set V = f~(-1)(0).Let B_εbe a ball inUof sufficiently small radius ε>0, centred at0 C~n. We show that f has an associated canonical pencil of real analytic hypersurfaces X_θ, with axis V,which leads to a fibratio n of the whole space(X B_ε)V over S~1.Its restriction to(S_)Vis the usual Milnor fibration =J/|f|,while its restriction to the Milnor tubef~(-1)( D_n) B_εis the Milnor-Le fibration of f. Each element of the pencil X_θmeets transversally the boundary sphereSe = B_ε,andthe intersection is the union of the link of f and two homeomorphic fibres of over antipodal points inthe circle. Furthermore, the space X obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibrebundle over RP~1with theX_θas fibres. These constructions work also, to some extent, for real analyticmap-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real andcomplex analytic singularities.