Refinements of Milnor's Fibration Theorem for Complex Singularities
| dc.creator | Cisneros-Molina, José-Luis | |
| dc.creator | Seade, Jose | |
| dc.creator | Snoussi, Jawad | |
| dc.date | 2007-12-14 | |
| dc.date | 2009-05-21 | |
| dc.date.accessioned | 2026-07-07T13:16:37Z | |
| dc.date.available | 2026-07-07T13:16:37Z | |
| dc.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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/0712.2440 | |
| dc.identifier | http://arxiv.org/abs/0712.2440 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230846 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Complex Variables | |
| dc.subject | 32S05, 32S55 | |
| dc.title | Refinements of Milnor's Fibration Theorem for Complex Singularities | |
| dc.type | text |