On holomorphic foliations admitting invariant CR manifolds

TL;DR

This paper investigates holomorphic foliations on complex manifolds, demonstrating that under certain positivity conditions of the normal bundle, such foliations cannot have compact invariant sets formed by smooth real hypersurfaces, contributing to the understanding of the exceptional minimal set conjecture.

Contribution

It establishes a new non-existence result for compact invariant sets in holomorphic foliations with Griffiths positive normal bundle, advancing the theory of invariant CR manifolds.

Findings

No compact invariant sets as complete intersections of smooth real hypersurfaces under Griffiths positivity.

Supports the exceptional minimal set conjecture in specific geometric contexts.

Provides conditions preventing certain invariant structures in holomorphic foliations.

Abstract

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.