Dynamical aspects of foliations with ample normal bundle

TL;DR

This paper proves a conjecture by Brunella that in certain complex manifolds, all leaves of a foliation with ample normal bundle accumulate to the foliation's singular set, revealing key dynamical behavior.

Contribution

It establishes the conjecture for compact complex manifolds of dimension at least three, demonstrating a fundamental property of foliations with ample normal bundles.

Findings

All leaves of the foliation accumulate to the singular set.

The result confirms Brunella's conjecture in the specified setting.

Provides insight into the dynamics of foliations with ample normal bundles.

Abstract

We prove the following result that was conjectured by Brunella: Let $X$ be a compact complex manifold of dimension $\geq 3$. Let $\mathcal{F}$ be a codimension one holomorphic foliation on $X$ with ample normal bundle. Then every leaf of $\mathcal{F}$ accumulates to the singular set of $\mathcal{F}$.