The homology groups of the links of quasi-ordinary singularities
type: Item
instance of: scholarly article
title: The homology groups of the links of quasi-ordinary singularities
zbMATH Open document ID: 0791.32016
DOI: 10.1007/BF03025740
author: Kyungho Oh
published in: Mathematische Zeitschrift
publication date: 1994-07-04T00:00:00Z
full work available at URL: https://eudml.org/doc/174548
review text: A \(d\)-dimensional complex analytic hypersurface germ \((X,x) \subset (\mathbb{C}^{d+1},0)\) is called a quasi-ordinary singularity if there is a branched covering \(\pi:(X,x) \to (\mathbb{C}^ d,0)\), which is unbranched outside the union of the coordinate hyperplanes in \(\mathbb{C}^ d\). The local ring \({\mathcal O}_{X,x}\) of germs of holomorphic functions is \(\mathbb{C}\{t_ 1,\dots,t_ d\} [\zeta]\), where \(\mathbb{C}\{t_ 1,\dots,t_ d\}\) is the convergent power series in \(d\) variables. \(\zeta\) is called a parametrization of \((X,x)\). Set \[ \begin{aligned} D_ i & = \{t \in \mathbb{C} \mid | t | \leq \varepsilon_ i\},\;\varepsilon_ i>0, \\ \partial {\mathbf D}^{d+1} & = \partial (D_ 1 \times \dots \times D_{d+1}), \\ L & = X \cap \partial {\mathbf D}^{d+1}. \end{aligned} \] The local topology of \((X,x) \subset (\mathbb{C}^{d+1},0)\) is determined by the pair \((\partial {\mathbf D}^{d+1},L)\) and \(L\) is called the link of the singularity \((X,x)\). The collection \({\mathcal U} = \{U_ i=\pi^{-1}(V_ i)\}\), where \(V_ i=(D_ 1 \times \dots \times D_{i-1} \times D^*_ i \times D_{i+1} \times \dots \times D_ d) \cap \partial {\mathbf D}^ d\), forms an open covering of \(L\). The Čech resolution of singular chain complexes of \(L\) with respect to the covering \({\mathcal U}\) gives rise to a spectral sequence: \[ E^ 2_{p,q} (L,{\mathcal U})=H^ \delta_ p \bigl( \bigoplus_{i_ 0< \cdots < i_ p} H_ q(U_{i_ 0 \dots i_ p},\mathbb{Z}) \bigr) \Rightarrow H_{p+q} (L,\mathbb{Z}), \] where \(U_{i_ 0 \dots i_ p}=U_{i_ 0} \cap \cdots \cap U_{i_ p}\). Now the map \(\pi\) induces a morphism \(\pi_ *\) between this spectral sequence and the following spectral sequence: \[ E^ 2_{p,q} (\partial {\mathbf D}^ d,{\mathcal V})=H^ \delta_ p \bigl( \bigoplus_{i_ 0<\cdots<i_ p} H_ q(V_{i_ 0\dots i_ p}, \mathbb{Z}) \bigr) \Rightarrow H_{p+q} (\partial {\mathbf D}^ d,\mathbb{Z}), \] where \({\mathcal V}=\{V_ i\}\) and \(V_{i_ 0 \dots i_ p}=V_{i_ 0} \cap \cdots \cap V_{i_ p}\). Our p…
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32B10
zbMATH DE Number: 561309
zbMATH Keywords: Galois group
zbMATH Keywords: quasi-ordinary singularity
zbMATH Keywords: spectral seuqence
zbMATH Keywords: link
MaRDI profile type: Publication
cites work: Q3961493
cites work: Local homological properties of analytic sets
cites work: Q5643279
cites work: Topological invariants of quasi-ordinary singularities
cites work: Q3807341
cites work: Q3671296
cites work: Q5678200
OpenAlex ID: W2031750250
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description: scientific article; zbMATH DE number 561309