A lower bound for the Milnor number of vector fields
Algebraic Geometry
2026-02-11 v1 Complex Variables
Differential Geometry
Dynamical Systems
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.
Cite
@article{arxiv.2602.10047,
title = {A lower bound for the Milnor number of vector fields},
author = {Maurício Corrêa and Gilcione Nonato Costa and Alejandra Salamanca Russi},
journal= {arXiv preprint arXiv:2602.10047},
year = {2026}
}
Comments
21 pages