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On the links of simple singularities, simple elliptic singularities and cusp singularities. (English) Zbl 1322.32021

Summary: This is a survey article about the study of the links of some complex hypersurface singularities in \(\mathbb{C}^3\). We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group \(\mathrm{SU}(2)\), \(\mathrm{Nil}^3\) or \(\mathrm{Sol}^3\), respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of \(\mathrm{S}^5\). Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in \(\mathrm{S}^5\) as codimension two contact submanifolds.

MSC:

32S25 Complex surface and hypersurface singularities
14J17 Singularities of surfaces or higher-dimensional varieties
57R17 Symplectic and contact topology in high or arbitrary dimension
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