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

2024/09/02 by Pedro Barbosa, Barbosa, Pedro, Arturo Fernández-Pérez +3
Mathematics · #Meromorphic and Entire Functions #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2409.01237

Abstract

We introduce the notion of the 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 ω with respect to a complex analytic hypersurface X with an isolated singularity can be expressed in terms of the Ebeling--Gusein-Zade index of ω along X, the Milnor number of ω and the 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 ω along X. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two.

Related

• The Bruce-Roberts Tjurina number of holomorphic 1-forms along complex analytic varieties
• Homological index for 1-forms and a Milnor number for isolated singularities
• The relative Bruce-Roberts number of a function on a hypersurface
• Proportionality of Indices of 1-Forms on Singular Varieties
• The Bruce-Roberts number of a function on a hypersurface with isolated\n singularity