Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous singularities
Let \(f(x_0, \dots, x_n) = 0\) be a germ of analytic hypersurface in \(\mathbb{C}^{ n + 1}\) with isolated singularity at the origin. The finite dimensional \(\mathbb{C}\)-algebra \(A = \mathbb{C} \{z_0, \dots, z_n\}/(f, \partial f/z_0, \dots, \partial f/z_n)\) is called the moduli algebra of the singularity: by a result of \textit{J. N. Mather} and \textit{S. Yau} [Invent. Math. 69, 243-251 (1992; Zbl 0499.32008)] two isolated singularities are analytically isomorphic if and only if their moduli algebras are isomorphic. To \(A\) there is associated the Lie algebra \(L\) of its derivations. If \(f\) is quasihomogeneous, then both \(A\) and \(L\) are naturally graded. In the present paper it is proven that in such a case all homogeneous pieces of negative degree of \(L\) are zero. This leads to a criterion characterizing quasihomogeneous singularities.
scientific article; zbMATH DE number 763192