A $6$-dimensional simply connected complex and symplectic manifold with   no K\"ahler metric
  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. 
Giovanni Bazzoni, Marisa Fern\'andez and Vicente Mu\~noz
math.SG math.AG math.DG