Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X , we show that under certain conditions on the singularity type of the pair ( f , X ), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair ( f , X ), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.