Methods and Applications of Analysis

Volume 21 (2014)

Number 3

Special issue dedicated to the 60th birthday of Stephen S.-T. Yau: Part I

Guest editors: John Erik Fornæss, Norwegian University of Science and Technology; Xiaojun Huang, Rutgers University; Song-Ying Li, University of California, Irvine; Yat Sun Poon, University of California, Riverside; Wing Shing Wong, The Chinese University of Hong Kong; and Zhouping Xin, The Institute of Mathematical Sciences, CUHK.

On two methods for reconstructing homogeneous hypersurface singularities from their Milnor algebras

Pages: 391 – 406

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n3.a8

Author

A. V. Isaev (Department of Mathematics, Australian National University, Canberra, Australia)

Abstract

By the well-known Mather-Yau theorem, a complex hypersurface germ $\mathcal{V}$ with isolated singularity is fully determined by its moduli algebra $\mathcal{A(V)}$. The proof of this theorem does not provide an explicit procedure for recovering $\mathcal{V}$ from $\mathcal{A(V)}$, and finding such a procedure is a long-standing open problem. In the present paper we survey and compare two recently proposed methods for reconstructing $\mathcal{V}$ from $\mathcal{A(V)}$ up to biholomorphic equivalence under the assumption that the singularity of $\mathcal{V}$ is homogeneous (in which case $\mathcal{A(V)}$ coincides with the Milnor algebra of $\mathcal{V}$). As part of our discussion of one of the methods, we give a characterization of the algebras arising from finite polynomial maps with homogeneous components of equal degrees.

Keywords

isolated hypersurface singularities, Milnor algebras, Mather-Yau theorem

2010 Mathematics Subject Classification

13H10, 32S25

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