On the normal bundle of Levi-flat real hypersurfaces

TL;DR

This paper proves that the normal bundle of Levi-flat hypersurfaces in complex manifolds cannot have a Hermitian metric with positive curvature along the leaves, extending previous results by Brunella.

Contribution

It generalizes Brunella's result to higher-dimensional complex manifolds, showing the non-existence of positively curved Hermitian metrics on the normal bundle of Levi-flat hypersurfaces.

Findings

Normal bundle of Levi-flat hypersurfaces cannot have positive curvature.

Generalization of Brunella's result to higher dimensions.

Supports the understanding of Levi-flat hypersurface geometry.

Abstract

Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.