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On the links of simple singularities, simple elliptic singularities and cusp singularities

Kasuya N.

Demonstratio Mathematica

2015

Vol. 48, nr 2

289--312

This is a survey article about the study of the links of some complex hypersurface singularities in C3. 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 SU (2), Nil3 or Sol3, 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 S5. 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 S5 as codimension two contact submanifolds.