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

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