Examples of application of nil-polynomials to the biholomorphic equivalence problem for isolated hypersurface singularities. (English) Zbl 1297.32018

Isaev, A. V.

Bull. Inst. Math., Acad. Sin. (N.S.) 8, No. 2, 193-217 (2013).

J. N. Mather and S. S. T. Yau [Invent. Math. 69, 243–251 (1982; Zbl 0499.32008)] proved that two complex hypersurface singularities are biholomorphically equivalent if their moduli algebras are isomorphic. In [G. Fels et al., J. Geom. Anal. 21, No. 3, 767–782 (2011; Zbl 1274.32018)] the authors showed that the equivalence problem for quasi-homogeneous hypersurface singularities can be reduced to the linear equivalence problem for nil-polynomials arising from the moduli algebras. In the paper under review, using the nil-polynomials, the author provides an explicit solution to the equivalence problem for hypersurface simple elliptic singularities, and also discusses the equivalence problem for a family of plane curve singularities \(x^n+tx^{n-1}y+y^n=0\) (\(t\in \mathbb C\)).

Reviewer: Tomohiro Okuma (Yamagata)

MSC:

32S25	Complex surface and hypersurface singularities
13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Keywords:

homogeneous singularity; isolated hypersurface singularity; simple elliptic singularity; moduli algebra