Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras

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 ${\mathcal V}$.