On the Milnor fibres of cyclic quotient singularities
On the Milnor fibres of cyclic quotient singularities
复制标题
关于循环商奇点的 Milnor 纤维
DOI：
10.1112/plms/pdq007
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发表时间：
2008
影响因子：
1.8
通讯作者：
Patrick Popescu
中科院分区：
文献类型：
--
作者：
A. Némethi;Patrick Popescu
循环商奇点 ρ, q 的有向链接与透镜空间 L(p, q) 保持方向微分同胚，并带有标准接触结构 ψst。 Lisca 将 (L(p, q), ψst) 的 Stein 填充分类为微分同胚，并推测它们通过显式映射双射对应于与 ?p、q 变形的约简微宇宙空间的不可约分量（所有这些都是平滑分量）相关的 Milnor 纤维。我们使用 Christophersen 和 Stevens 给出的平滑方程证明了这个猜想。此外，根据 de Jong 和 van Straten 对 Milnor 纤维的不同描述，我们也将这些纤维与 Lisca 填充物进行了规范的识别。使用这些和新引入的与透镜空间相关的附加结构（顺序），我们证明上述 Milnor 纤维是成对非微分同胚的（通过保留方向和顺序的微分同胚）。这也意味着德容和范斯特拉滕以与克里斯托弗森和史蒂文斯相同的方式参数化减少的微型变形空间的组成部分。
The oriented link of the cyclic quotient singularity ?p, q is orientation‐preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ξst. Lisca classified the Stein fillings of (L(p, q), ξst) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of ?p, q. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca's fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non‐diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.