Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A lower bound for the Milnor number of vector fields

Maurício Corrêa, Gilcione Nonato Costa, Alejandra Salamanca Russi

TL;DR

The paper addresses the problem of lower-bounding the limiting Milnor (or Poincaré–Hopf) contribution from a non-isolated, positive-dimensional singular component $W$ of a holomorphic vector field under holomorphic perturbations. It develops global and local formulas that isolate the $W$-localized contribution, using the embedded scheme structure along $W$, and introduces a parameter-dependent $\mathcal{K}^{(d)}$-determinacy framework to bound this contribution. A key outcome is a sharp lower bound $\mu(\mathcal{F}_t, W) \ge N(\mathcal{F},A_W)$, with a decomposition $\mu(\mathcal{F},W) = N(\mathcal{F},A_W) - \nu(\mathcal{F},W,\varphi_0)$, and explicit examples demonstrating both optimality and the redistribution of singular mass between a fixed neighborhood of $W$ and infinity under projective compactifications. The results connect to polar-type invariants and provide computable, deformation-invariant data that clarifies how positive-dimensional singularities contribute to index counts in both affine and projective settings, including cases where the contribution can be forced to vanish or become unbounded.

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.

A lower bound for the Milnor number of vector fields

TL;DR

The paper addresses the problem of lower-bounding the limiting Milnor (or Poincaré–Hopf) contribution from a non-isolated, positive-dimensional singular component of a holomorphic vector field under holomorphic perturbations. It develops global and local formulas that isolate the -localized contribution, using the embedded scheme structure along , and introduces a parameter-dependent -determinacy framework to bound this contribution. A key outcome is a sharp lower bound , with a decomposition , and explicit examples demonstrating both optimality and the redistribution of singular mass between a fixed neighborhood of and infinity under projective compactifications. The results connect to polar-type invariants and provide computable, deformation-invariant data that clarifies how positive-dimensional singularities contribute to index counts in both affine and projective settings, including cases where the contribution can be forced to vanish or become unbounded.

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.
Paper Structure (6 sections, 8 theorems, 123 equations)

This paper contains 6 sections, 8 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{K}$-equivalence
  4. Holomorphic vector fields with non-isolated singularities
  5. Examples
  6. Acknowledgments.

Key Result

Theorem 1.1

Let $X = X_1\frac{\partial}{\partial z_1}+\cdots+X_n\frac{\partial}{\partial z_n}$ be a germ of a holomorphic vector field at $0 \in \mathbb{C}^n$ such that $0$ is an isolated zero of $X$, and let $k$ be the degree of $\mathcal{K}$-determinacy of $X$. Then:

Theorems & Definitions (18)

  • Theorem 1.1: MGS
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3: Artin approximation, analytic version ArtinApprox2018
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • ...and 8 more