The relative Bruce-Roberts number of a function on a hypersurface

TL;DR

This paper establishes a formula linking the relative Bruce-Roberts number of a function on a hypersurface to classical invariants like the Milnor and Tjurina numbers, and explores properties of the associated logarithmic characteristic variety.

Contribution

It provides a new explicit relation for the relative Bruce-Roberts number and analyzes the Cohen-Macaulay property of the relative logarithmic characteristic variety.

Findings

$ ext{μ}_{BR}^{-}(f,X)$ equals the sum of the Milnor number of the fiber and the difference between Milnor and Tjurina numbers.

$ ext{μ}_{BR}(f,X)$ equals the sum of $ ext{μ}(f)$ and $ ext{μ}_{BR}^{-}(f,X)$.

The relative logarithmic characteristic variety $LC(X)^-$ is Cohen-Macaulay.

Abstract

We consider the relative Bruce-Roberts number $\mu_{BR}^{-}(f,X)$ of a function on an isolated hypersurface singularity $(X,0)$. We show that $\mu_{BR}^{-}(f,X)$ is equal to the sum of the Milnor number of the fibre $\mu(f^{-1}(0)\cap X,0)$ plus the difference $\mu(X,0)-\tau(X,0)$ between the Milnor and the Tjurina numbers of $(X,0)$. As an application, we show that the usual Bruce-Roberts number $\mu_{BR}(f,X)$ is equal to $\mu(f)+\mu_{BR}^{-}(f,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^-$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen-Macaulay.