The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties

TL;DR

This paper introduces the Bruce-Roberts number for holomorphic 1-forms on complex varieties, relating it to indices and classical invariants, and explores its applications to foliations in complex dimension two.

Contribution

It defines the Bruce-Roberts number for 1-forms relative to complex varieties and links it to existing invariants like the Milnor and Tjurina numbers, extending known formulas.

Findings

Expressed Bruce-Roberts number in terms of Ebeling-Gusein-Zade index, Milnor number, and Tjurina number.

Connected Bruce-Roberts number with radial index and local Euler obstruction.

Applied results to holomorphic foliations in complex dimension two.

Abstract

We introduce the notion of the \textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $\omega$ with respect to a complex analytic hypersurface $X$ with an isolated singularity can be expressed in terms of the \textit{Ebeling--Gusein-Zade index} of $\omega$ along $X$, the \textit{Milnor number} of $\omega$ and the \textit{Tjurina number} of $X$. This result allows us to recover known formulas for the Bruce-Roberts number of a holomorphic function along $X$ and to establish connections between this number, the radial index, and the local Euler obstruction of $\omega$ along $X$. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two.