A lower bound for the Milnor number of vector fields

TL;DR

This paper establishes lower bounds for the Milnor number of holomorphic vector fields with specific singularities, providing formulas and examples to understand how singularities behave under perturbations.

Contribution

It introduces new formulas for Milnor number contributions along singular components and demonstrates sharp lower bounds under perturbations.

Findings

Derived explicit formulas for Milnor number contributions.

Proved sharp lower bounds for perturbations.

Provided examples illustrating optimality and singularity redistribution.

Abstract

We study holomorphic vector fields whose singular locus contains a local complete intersection smooth positive-dimensional component. We prove global and local formulas expressing the limiting Milnor/Poincare-Hopf contribution along such a component in terms of its embedded scheme structure, and we obtain sharp lower bounds for this contribution under holomorphic perturbations. We provide explicit families show optimality and illustrate how singularities may redistribute between a fixed neighborhood of the component and the part at infinity in projective compactifications.