A lower bound for the Milnor number of vector fields

Maurício Corrêa; Gilcione Nonato Costa; Alejandra Salamanca Russi

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arXiv · 2602.10047

A lower bound for the Milnor number of vector fields

Maurício Corrêa·Gilcione Nonato Costa·Alejandra Salamanca Russi

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Abstract

We study holomorphic vector fields whose singular locus contains a smooth positive-dimensional local complete-intersection component. We obtain an exact global formula for its Milnor-Poincar\'e-Hopf contribution and prove a sharp local lower bound, valid under holomorphic perturbations, in terms of the embedded contribution associated with the component. As an application to gradient vector fields, we derive a lower bound for the total Milnor number of the isolated critical points arising near a smooth positive-dimensional critical component; for a morsification, this becomes a lower bound for the number of Morse critical points converging to that component. Explicit families show that the bounds are sharp and exhibit the redistribution of singularities between a fixed neighbourhood of the component and the hyperplane at infinity in projective compactifications.

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Maurício Corrêa, Gilcione Nonato Costa, Alejandra Salamanca Russi. 2026-02-10. A lower bound for the Milnor number of vector fields. https://arxiv.org/abs/2602.10047

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