Micro-local characterization of quasi-homogeneous singularities

A moduli algebra A(V) of hypersurface singularity (V, 0) is a finite dimensional C-algebra. In 1982, Mather and Yau proved that two germs of complex analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebra are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra. In this paper we prove that (V, 0) admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra. In 1983, Yau introduced a finite dimensional Lie algebra L(V) to an isolated hypersurface singularity (V, 0). L(V) is defined to be the algebra of derivations of the moduli algebra A(V) and is finite dimensional. We prove that (V, 0) is quasi-homogeneous singularity if (1) L(V) is isomorphic to a nonnegatively graded Lie algebra without center, (2) There exists E in L(V) of degree zero such that [E, D_i] = i D_i for any D_i in L(V) of degree i, and (3) For any element a ∈ m - m² where m is the maximal ideal of A(V), aE is not in degree zero part of L(V).