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

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.