摘要翻译：
设$X$为$\mathbb{C}^n$中原点$\下划线{0}$的开邻域$u$的分析子集。设$F\冒号（X,\下划线{0}）\到(\mathbb{C},0)$为全纯，并设置$V=f^{-1}(0)$。设$\b_\epsilon$是一个位于$u$中的球，半径足够小$\epsilon>0$，以$\下划线{0}\in\mathbb{C}^n$为中心。我们证明$F$有一个与轴$V$相关联的实解析超曲面的正则束$X_\theta$,它导致整个空间$(X\cap\mathbb{B}_psilon)\set-v$在$\mathbb{S}^1$上的纤维化$\phi$。它对$(X\cap\mathbb{S}_\epsilon)\setminus v$的限制是通常的米尔诺纤维$\phi=\frac{f}{f}$,而对米尔诺管$f^{-1}(\partial\d_\eta)\cap\mathbb{B}_\epsilon$的限制是$f$的米尔诺-L\\e纤维。铅笔的每一个元素与边界球面$\mathbb{S}_\epsilon=\partial\b_\epsilon$横向相交，其交点是$f$与两个$\phi$同胚纤维在圆内对角点上的并。此外，由理想的$(Re(f)，Im(f))$的实际爆破得到的空间${\tilde X}$是$\mathbb{R}\mathbb{P}^1$上的纤维束，其中$X\theta$为纤维。这些构造在某种程度上也适用于实解析映射芽，并给我们提供了实解析奇点和复解析奇点之间关于Milnor纤维的区别的清晰图像。
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英文标题：
《Refinements of Milnor's Fibration Theorem for Complex Singularities》
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作者：
Jos\'e-Luis Cisneros-Molina, Jose Seade and Jawad Snoussi
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0712.2440