A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

Abstract: We construct a simply connected compact manifold which has complex and symplectic structures but does not admit K\"ahler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.