Non existence of Levi flat hypersurfaces with positive normal bundle in compact K\"ahler manifolds of dimension at least 3

TL;DR

This paper proves that in compact Kähler manifolds of dimension at least 3, Levi flat hypersurfaces cannot have a normal bundle with a Hermitian metric of positive curvature along the leaves, confirming a conjecture by Marco Brunella.

Contribution

It establishes the non-existence of Levi flat hypersurfaces with positively curved normal bundles in higher-dimensional compact Kähler manifolds, resolving a longstanding conjecture.

Findings

Normal bundle to Levi foliation cannot have positive curvature

No Levi flat hypersurfaces with positive normal bundle exist in these manifolds

Confirms Brunella's conjecture in higher dimensions

Abstract

We prove that the normal bundle to the Levi foliation of a smooth Levi flat hypersurface does not admit a Hermitian metric with positive curvature along the leaves in compact K\"ahler manifolds of dimension at least 3. This represents an answer to a conjecture of Marco Brunella.