Mixed Bruce-Roberts numbers

TL;DR

This paper extends Bruce-Roberts numbers to pairs of complex analytic varieties and functions, analyzing their fundamental properties to deepen understanding of singularity invariants in complex geometry.

Contribution

It introduces a generalized framework for Bruce-Roberts numbers for pairs of varieties and functions, expanding their theoretical foundation.

Findings

Extended $oldsymbol{ ext{μ}^*}$-sequence and Tjurina number concepts.

Analyzed fundamental properties of these extended numbers.

Provided insights into singularity invariants in complex analytic geometry.

Abstract

We extend the notion of $\mu^*$-sequence and Tjurina number of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at $0$ of a complex analytic variety $X\subseteq \mathbb C^n$ and a finitely $\mathcal R(X)$-determined analytic function germ $f:(\mathbb C^n,0)\to (\mathbb C,0)$. We analyze some fundamental properties of these numbers.