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Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras


Authors: A. V. Isaev and N. G. Kruzhilin
Journal: Proc. Amer. Math. Soc. 142 (2014), 581-590
MSC (2010): Primary 32S25, 13H10
Published electronically: October 31, 2013
MathSciNet review: 3133999
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Abstract: By the Mather-Yau theorem, a complex hypersurface germ $ {\mathcal V}$ with isolated singularity is completely determined by its moduli algebra $ {\mathcal A}({\mathcal V})$. The proof of the theorem does not provide an explicit procedure for recovering $ {\mathcal V}$ from $ {\mathcal A}({\mathcal V})$, and finding such a procedure is a long-standing open problem. In this paper, we present an explicit way for reconstructing $ {\mathcal V}$ from $ {\mathcal A}({\mathcal V})$ up to biholomorphic equivalence under the assumption that the singularity of $ {\mathcal V}$ is homogeneous, in which case $ {\mathcal A}({\mathcal V})$ coincides with the Milnor algebra of $ {\mathcal V}$.


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Additional Information

A. V. Isaev
Affiliation: Department of Mathematics, The Australian National University, Canberra, ACT 0200, Australia
Email: alexander.isaev@anu.edu.au

N. G. Kruzhilin
Affiliation: Department of Complex Analysis, Steklov Mathematical Institute, 8 Gubkina Street, Moscow GSP-1 119991, Russia
Email: kruzhil@mi.ras.ru

DOI: https://doi.org/10.1090/S0002-9939-2013-11822-8
Received by editor(s): March 25, 2012
Published electronically: October 31, 2013
Communicated by: Franc Forstneric
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.