# Dicritical nilpotent holomorphic foliations

## TL;DR

This paper investigates the properties of singularities in three-dimensional holomorphic foliations that are pull-backs of dicritical foliations in two dimensions, focusing on first integrals and dicriticalness.

## Contribution

It extends the study of dicritical foliations and their singularities from two to three dimensions, analyzing the existence of first integrals and the dicritical nature.

## Key findings

- Existence criteria for holomorphic and meromorphic first integrals.
- Characterization of dicriticalness in three-dimensional pull-back foliations.
- Adaptation of two-dimensional methods to three-dimensional cases.

## Abstract

We study in this paper several properties concerning singularities of foliations in $(\mathbb{C}^3,\mathbf{0})$ that are pull-back of dicritical foliations in $(\mathbb{C}^2,\mathbf{0})$. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in $(\mathbb{C}^2,\mathbf{0})$, the adaptations are not straightforward.