Unique ergodicity for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with an invariant plane

TL;DR

This paper establishes a unique ergodicity theorem for certain singular holomorphic foliations in complex projective 3-space, extending previous results from lower dimensions using dynamical methods and integrability estimates.

Contribution

It generalizes the concept of unique ergodicity to singular foliations in -space, adapting techniques from lower-dimensional cases and singular dynamics.

Findings

Proves unique ergodicity for foliations with hyperbolic singularities and invariant planes.

Extends dynamical methods to singular holomorphic foliations in -space.

Utilizes integrability estimates to handle singularities in the proof.

Abstract

We prove a unique ergodicity theorem for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with hyperbolic singularities and with an invariant plane with no foliation cycle, in analogy with a result of Dinh-Sibony concerning unique ergodicity for foliations of $\mathbb{P}^2(\mathbb{C})$ with an invariant line. The proof is dynamical in nature and adapts the work of Deroin-Kleptsyn to a singular context, using the fundamental integrability estimate of Nguy\^en.