A lower bound for the Milnor number of vector fields

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.

💡 Research Summary

The paper investigates holomorphic vector fields (or one‑dimensional holomorphic foliations) whose singular set contains a smooth positive‑dimensional component W that is a local complete intersection of codimension d ≥ 2. In the non‑isolated setting the usual Milnor number is not defined, because singularities are not isolated. The authors therefore introduce a “limiting Milnor contribution” along W, \