A lower bound for the Milnor number of vector fields

Technical Summary: A Lower Bound for the Milnor Number of Vector Fields

Problem Statement
The paper addresses the behavior of holomorphic vector fields $X$ on $\mathbb{C}^n$ whose singular locus, $\text{Sing}(X)$, contains a smooth, positive-dimensional local complete-intersection subvariety $W$ of codimension $d \geq 2$. While the theory of isolated singularities is well-established (notably through Soares' work on $\mathcal{K}$-equivalence and finite determinacy), the case of non-isolated singular sets presents distinct challenges. Under a holomorphic deformation $\{X_t\}$, isolated singularities may emerge from $W$, remain in a fixed neighborhood, or converge to $W$ as the parameter $t \to 0$. Furthermore, upon projective compactification, part of the total index may concentrate at the hyperplane at infinity.

The central question is whether a uniform lower bound exists for the total Milnor number (or Poincaré–Hopf index) of the isolated singularities converging to $W$ under any admissible holomorphic perturbation. Specifically, does there exist a quantity $\mu_0(X, W)$ such that for every deformation $\{X_t\}$ with finite singular sets for $t \neq 0$, the limit of the sum of Milnor numbers of points converging to $W$ satisfies $\mu(X_t, W) \geq \mu_0(X, W)$?

Methodology
The authors employ a combination of deformation theory, intersection theory, and the study of holomorphic foliations.

1. Deformation and Perturbation: The study focuses on "embedded singular points," defined as isolated singularities produced by sufficiently small admissible deformations that converge to $W$. This aligns with the deformation theory of Baum–Bott residues developed by Bracci and Suwa.
2. $\mathcal{K}$-Equivalence and Determinacy: The paper utilizes Mather's theory of $\mathcal{K}$-determinacy. For isolated singularities, finite determinacy allows reduction to a finite jet. For positive-dimensional components, the authors extend this via parameter-dependent $\mathcal{K}(d)$-equivalence, analyzing equidimensional slices of the map defining $W$.
3. Projective Compactification and Blow-ups: The analysis involves blowing up $\mathbb{P}^n$ along $W$ to study the behavior of the foliation along the exceptional divisor. The concept of "dicritical" versus "non-dicritical" components is crucial for determining the order of vanishing along the exceptional divisor.
4. Polynomial Approximation: Theorem 3.1 establishes that any holomorphic foliation with a smooth complete intersection singular set can be approximated by polynomial foliations, allowing the use of algebraic techniques to study the limiting behavior of indices.
5. Gradient Fields: The results are applied to gradient vector fields $\nabla f$ of holomorphic function germs, linking the Milnor number of the vector field to the Milnor number of the function at critical points.

Key Contributions and Results

• Exact Global Formula (Theorem 1.2): The authors provide an exact global formula for the Milnor number of a holomorphic foliation on $\mathbb{P}^n$ ($n \geq 3$) with a singular locus consisting of smooth, non-dicritical complete intersection components $W_j$ and isolated points. The formula relates the sum of indices at isolated points to a residual term $\nu(\mathcal{F}, W_j, \varphi_0)$ and the number of embedded points $N(\mathcal{F}, A_{W_j})$:
  $$\sum \mu(\mathcal{F}, p_i) = \sum k^i + \sum \nu(\mathcal{F}, W_j, \varphi_0) - \sum N(\mathcal{F}, A_{W_j})$$
  This identifies the contribution supported on positive-dimensional components.

• Sharp Local Lower Bound (Theorem 1.3): The primary result is a local lower bound for the Milnor number of singularities converging to $W$. For a germ of a holomorphic foliation $\mathcal{F}$ on $\mathbb{C}^n$ with a smooth complete intersection $W \subset \text{Sing}(\mathcal{F})$, the paper proves:
  $$\mu(\mathcal{F}_t, W) \geq N(\mathcal{F}, A_W)$$
  where $N(\mathcal{F}, A_W)$ is the number of embedded closed points associated with $W$ (counted with multiplicity) in the perturbative sense. This bound is sharp; the paper demonstrates that for "totally simple" foliations (where the Jacobian matrix has a non-vanishing minor along $W$), there exists a perturbation where the limit of the sum of Milnor numbers is exactly zero.

• Application to Gradient Fields (Corollary 1.4): Applying the vector field results to gradient fields $\nabla f$, the authors derive a lower bound for the total Milnor number of isolated critical points arising near a smooth positive-dimensional critical component $W$. In the case of a morsification (where the deformation yields non-degenerate critical points), this becomes a lower bound for the number of Morse critical points converging to $W$.

• Explicit Families and Sharpness (Section 4): The authors construct explicit families of vector fields to demonstrate the sharpness of the bounds. These examples illustrate:

  • The redistribution of singularities between a fixed affine neighborhood and the hyperplane at infinity.
  • Cases where the contribution at infinity can be arbitrarily large (via polynomial truncations of transcendental coefficients), showing no uniform upper bound exists for the contribution at infinity in general classes.
  • Scenarios where the lower bound is achieved (e.g., when the perturbation eliminates singularities near $W$ entirely in the totally simple case).

Significance and Claims
The paper claims to resolve the question of whether a lower bound exists for the Milnor number of singularities converging to a positive-dimensional component. It establishes that the "embedded contribution" (the number of embedded points $N(\mathcal{F}, A_W)$) serves as this sharp lower bound.

The authors position their work as complementary to existing theories involving Lê cycles, Lê numbers, and polar multiplicities (Teissier, Gaffney). While those theories describe the splitting of critical loci via Chern-theoretic invariants, this paper isolates a specific lower bound determined by the embedded contribution. The results clarify the behavior of singularities under deformation, distinguishing between those that remain near the component and those that migrate to infinity, and provide precise formulas for the total index in projective settings. The work extends the classical theory of isolated singularities (Soares, Esteves, Vainsencher) to the non-isolated case, offering a unified framework for calculating Milnor numbers in the presence of positive-dimensional singular sets.