Refinements of Milnor's Fibration Theorem for Complex Singularities

Cisneros-Molina, José-Luis; Seade, Jose; Snoussi, Jawad

Refinements of Milnor's Fibration Theorem for Complex Singularities

Authors

Cisneros-Molina, José-Luis

Seade, Jose

Snoussi, Jawad

Description

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_ε$ be a ball in $U$ of sufficiently small radius $ε>0$, centred at $\underline{0}\in\mathbb{C}^n$. 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 $(X \cap \mathbb{B}_ε) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_ε) \setminus V$ is the usual Milnor fibration $ϕ= \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_η) \cap \mathbb{B}_ε$ is the Milnor-Lê fibration of $f$. Each element of the pencil $X_θ$ meets transversally the boundary sphere $\mathbb{S}_ε= \partial \B_ε$, 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 ${\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\mathbb{R} \mathbb{P}^1$ 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.
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