Demonstratio Mathematica (Jun 2015)

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

  • Kasuya Naohik

DOI
https://doi.org/10.1515/dema-2015-0021
Journal volume & issue
Vol. 48, no. 2
pp. 289 – 312

Abstract

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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.

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