Generalized Poincar\'e-Dulac singularities of holomorphic foliations

TL;DR

This paper classifies a specific type of nilpotent singularities in holomorphic foliations in complex two-space, revealing their rigidity and dependence on holonomy, thus advancing understanding of their local behavior.

Contribution

It provides a new analytic classification for Poincaré-Dulac type singularities in holomorphic foliations, highlighting their formal analytic rigidity and holonomy-based invariants.

Findings

Singularities exhibit formal analytic rigidity.

Classification based on holonomy of exceptional divisor component.

Advances understanding of local behavior of holomorphic foliations.

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 Poincar\'e-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.