  Given a surjective holomorphic map between two compact Kaehler manifolds, we
investigate the positivity properties of relative adjoint transcendental
classes which are inherited from similar fiberwise properties. As a
consequence, we solve a problem proposed by J.-P. Demailly, T. Peternell and M.
Schneider concerning the Albanese morphism of Kaehler manifolds with nef
anticanonical class.
