Refinements of Milnor's Fibration Theorem for Complex Singularities

Abstract

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 $\B_\epsilon$ be a ball in $U$ of sufficiently small radius $\epsilon>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_\theta$, with axis $V$, which leads to a fibration $\Phi$ of the whole space $(X \cap \mathbb{B}_\epsilon) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_\epsilon) \setminus V$ is the usual Milnor fibration $\phi = \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_\eta) \cap \mathbb{B}_\epsilon$ is the Milnor-L\^e fibration of $f$. Each element of the pencil $X_\theta$ meets transversally the boundary sphere $\mathbb{S}_\epsilon = \partial \B_\epsilon$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $\phi$ 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 $X_\theta$ 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.