A lower bound for the Milnor number of vector fields

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-Poincaré-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.