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Generalized Poincaré-Dulac singularities of holomorphic foliations

Percy Fernández Sánchez, Jorge Mozo Fernández

Abstract

In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in $(\mathbb{C}^2,0)$, those exhibiting a Poincaré-Dulac type singularity in their reduction process. This analytic classification is based in the holonomy of a certain component of the exceptional divisor. Finally, as a consequence, we show that these singularities exhibit a formal analytic rigidity.

Generalized Poincaré-Dulac singularities of holomorphic foliations

Abstract

In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in , those exhibiting a Poincaré-Dulac type singularity in their reduction process. This analytic classification is based in the holonomy of a certain component of the exceptional divisor. Finally, as a consequence, we show that these singularities exhibit a formal analytic rigidity.

Paper Structure

This paper contains 6 sections, 2 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Nilpotent singularities in dimension two. Generalized Poincaré-Dulac singularities.
  3. Holonomy and analytic classification
  4. Structure of the holonomy group
  5. Previous results and proof of the main Theorem
  6. Realizability and Rigidity

Key Result

Theorem 1

Let ${\mathcal{F}}_1$, ${\mathcal{F}}_2$ be two germs of generalized Poincaré-Dulac holomorphic foliations as in formaprenormal, formally equivalent, with $n=2p$. Assume that $H_i$ is the holonomy group of the $p$th component of the exceptional divisor obtained during the process of reduction of sin

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Theorem 1
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['teoremaprincipal']}