Derivations of the moduli algebras of weighted homogeneous hypersurface singularities

Let R=C[x1,x2,⋯,zn]/(f)R=C[x1,x2,⋯,zn]/(f) where f is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then R  , and Der(R)Der(R), the Lie algebra of derivations on R  , are graded. It is well-known that Der(R)Der(R) has no negatively graded component [10]. J. Wahl conjectured that the above fact is still true in higher codimensional case provided that R=C[x1,x2,⋯,xn]/(f1,f2,⋯,fm)R=C[x1,x2,⋯,xn]/(f1,f2,⋯,fm) is an isolated, normal and complete intersection singularity and f1,f2,⋯,fmf1,f2,⋯,fm are weighted homogeneous polynomials with the same weight type (w1,w2,⋯,wn)(w1,w2,⋯,wn). On the other hand the first author Yau conjectured that the moduli algebra A(V)=C[x1,x2,⋯,xn]/(∂f/∂x1,⋯,∂f/∂xn)A(V)=C[x1,x2,⋯,xn]/(∂f/∂x1,⋯,∂f/∂xn) has no negatively weighted derivations where f   is a weighted homogeneous polynomial defining an isolated singularity at the origin. Assuming this conjecture has a positive answer, he gave a characterization of weighted homogeneous hypersurface singularities only using the Lie algebra Der(A(V))Der(A(V)) of derivations on A(V)A(V). The conjecture of Yau can be thought as an Artinian analogue of J. Wahl's conjecture. For the low embedding dimension, the Yau conjecture has a positive answer. In this paper we prove this conjecture for any high-dimensional singularities under the condition that the lowest weight is bigger than or equal to half of the highest weight.