Bruce-Roberts Numbers and Quasihomogeneous Functions on Analytic Varieties

TL;DR

This paper extends classical results relating Milnor and Tjurina numbers to quasihomogeneity, to the setting of analytic varieties and functions with stratified isolated singularities, establishing conditions for their equality.

Contribution

It proves a relative version of Saito's theorem linking Bruce-Roberts numbers and quasihomogeneity for functions on analytic varieties.

Findings

Equality of relative Milnor and Tjurina numbers implies relative quasihomogeneity.

Under certain conditions, the pair (f,X) is quasihomogeneous if and only if their Bruce-Roberts numbers coincide.

The results generalize classical singularity theory to stratified analytic varieties.

Abstract

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.