2007/12/14 by José-Luis Cisneros-Molina, Cisneros-Molina, José-Luis, José Seade +4 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG #math.CV #msc:32S05 #msc:32S55
paper · pdf · doi:10.48550/arxiv.0712.2440
37 pages, LaTeX; slightly modified title and abstract, rewrote introduction, reorganized parts of the paper and references added; some errors have been fixed and some improved results added; some lemmas added and a proof extended. To appear in Advances in Mathematics
Let X be an analytic subset of an open neighbourhood U of the origin \underline0 in ℂn. Let f\colon (X,\underline0) → (ℂ,0) be holomorphic and set V =f-1(0). Let \Bε be a ball in U of sufficiently small radius ε>0, centred at \underline0∈ℂn. 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 ∩ \mathbbBε) ∖ V over \mathbbS1 . Its restriction to (X ∩ \mathbbSε) ∖ V is the usual Milnor fibration ϕ= (f)/(|f|), while its restriction to the Milnor tube f-1(∂ \Dη) ∩ \mathbbBε is the Milnor-Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere \mathbbSε= ∂ \Bε, and the intersection is the union of the link of f and two homeomorphic fibers 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 ℝ ℙ1 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.