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Global residue formula for logarithmic indices of foliations

Maurício Corrêa, Diogo da Silva Machado

Abstract

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invariant divisor. As an application, we provide a formula for the number of singularities in the complement of the invariant divisor on complex projective spaces. Finally, we obtain a Poincaré-Hopf type formula for singular normal projective varieties.

Global residue formula for logarithmic indices of foliations

Abstract

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invariant divisor. As an application, we provide a formula for the number of singularities in the complement of the invariant divisor on complex projective spaces. Finally, we obtain a Poincaré-Hopf type formula for singular normal projective varieties.

Paper Structure

This paper contains 9 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic forms and vector fields
  4. Singular one-dimensional holomorphic foliations
  5. Projective holomorphic foliations
  6. Logarithmic and homological indices
  7. Proof of Theorem \ref{['teo1_1']}
  8. Proof Corollary \ref{['cor-Pn']}
  9. Proof Corollary \ref{['singlogres']}

Key Result

Corollary 1.1

Let $\mathscr F$ be a one-dimensional foliation on $X$, with isolated singularities and logarithmic along a normal crossings divisor $D$.

Theorems & Definitions (10)

  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Lemma 3.6
  • Proposition 3.7