Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras

TL;DR

This paper provides an explicit method to reconstruct homogeneous isolated hypersurface singularities from their Milnor algebras, addressing a long-standing open problem in singularity theory.

Contribution

It introduces a constructive procedure for recovering the hypersurface from its Milnor algebra when the singularity is homogeneous.

Findings

Reconstruction method explicitly recovers the hypersurface from Milnor algebra.

Applicable to homogeneous isolated hypersurface singularities.

Advances understanding of the Mather-Yau theorem in explicit terms.

Abstract

By the Mather-Yau theorem, a complex hypersurface germ $V$ with isolated singularity is completely determined by its moduli algebra $A(V)$. The proof of the theorem does not provide an explicit procedure for recovering $V$ from $A(V)$, and finding such a procedure is a long-standing open problem. In this paper, we present an explicit way for reconstructing $V$ from $A(V)$ up to biholomorphic equivalence under the assumption that the singularity of $V$ is homogeneous, in which case $A(V)$ coincides with the Milnor algebra of $V$.