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A higher-dimensional Van den Essen type formula for projective foliations and applications

Maurício Corrêa, Gilcione Nonato Costa

TL;DR

This work extends Van den Essen's residue-type formula to higher dimensions, deriving a Milnor-number variation formula for projective foliations on $\mathbb{P}^n$ with non-isolated singularities supported along a smooth complete intersection $W$. By coupling deformation theory with Baum–Bott/Chern-class calculations, the authors relate $\mu(\mathcal{F},W)$ to invariants of $W$ and the vanishing order along exceptional divisors, obtaining a universal lower bound and a precise blow-up law. They then specialize to foliations on $\mathbb{P}^n$, giving explicit multi-step formulas for $\mu(\mathcal{F}_j,W_j)$ and a bound on how many blow-ups are needed to desingularize, with $N(\mathcal{F}_j,A_{W_j})$ tracking embedding points. As an application, the paper proves a Seidenberg-type desingularization: after finitely many blow-ups along curves homeomorphic to $W_0$, the foliation becomes elementary at generic points and admits a natural local normal form, providing a concrete desingularization bound in terms of topological and numerical invariants of $W_0$.

Abstract

Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ μ(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve.

A higher-dimensional Van den Essen type formula for projective foliations and applications

TL;DR

This work extends Van den Essen's residue-type formula to higher dimensions, deriving a Milnor-number variation formula for projective foliations on with non-isolated singularities supported along a smooth complete intersection . By coupling deformation theory with Baum–Bott/Chern-class calculations, the authors relate to invariants of and the vanishing order along exceptional divisors, obtaining a universal lower bound and a precise blow-up law. They then specialize to foliations on , giving explicit multi-step formulas for and a bound on how many blow-ups are needed to desingularize, with tracking embedding points. As an application, the paper proves a Seidenberg-type desingularization: after finitely many blow-ups along curves homeomorphic to , the foliation becomes elementary at generic points and admits a natural local normal form, providing a concrete desingularization bound in terms of topological and numerical invariants of .

Abstract

Let be a one-dimensional holomorphic foliation on such that , where is a smooth complete intersection variety. We determine and compute the variation of the Milnor number under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of . This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve.
Paper Structure (13 sections, 18 theorems, 258 equations)

This paper contains 13 sections, 18 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Holomorphic foliations with non-isolated singularities
  4. Chern classes
  5. Fundamental Lemma of deformation
  6. Embedded closed points
  7. Proof of Theorem \ref{['theorem1']}
  8. Holomorphic foliations on $\mathbb{P}^n$
  9. Proof of Theorem \ref{['theoremB']}
  10. An application: Seidenberg's Theorem for non-isolated singularities
  11. Maximum number of blowups for the desingularization
  12. Normal forms for non-isolated singularities
  13. Proof of Theorem \ref{['theoremA']}

Key Result

Theorem 1.1

Let ${\mathcal{F}}_0$ be a holomorphic foliation by curves on $\mathbb{P}^n$, with $n\geq 3$, of degree $k$. Suppose that the singular set of ${\mathcal{F}}_0$ is the disjoint union of a smooth scheme-theoretic complete intersection subvariety ${\bf W}_0$ of pure codimension $d\geq 2$, and closed po where $N({\mathcal{F}}_0,\mathcal{A}_{{\bf W}_0})$ is the number of embedding closed points associa

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 37 more