Non-existence of Negative Weight Derivations of the Moduli Algebras of Isolated Weighted Homogeneous Hypersurface Singularities

Mather and Yau proved that the analytic isomorphism class of an isolated complex hypersurface singularity is determined by its dimension and its moduli algebra. Yau’s conjecture asserts that the moduli algebra of an isolated weighted homogeneous hypersurface singularity over $\mathbb{C}$ has no nonzero derivation of negative degree. We prove this conjecture. The proof uses Varchenko\textquotesingle s theorem on the $\mu$-constant stratum of an isolated weighted homogeneous singularity, also stated by M. Saito.