Relative adjoint transcendental classes and Albanese map of compact Kahler manifolds with nef Ricci curvature

We establish here the surjectivity of the Albanese map corresponding to compact Kahler manifold whose anti-canonical bundle is nef. This answers a question raised by J.-P. Demailly, Th. Peternell and M. Schneider. One of the main techniques we are using is the positivity of the relative canonical bundle associated to Kahler fiber spaces, obtained via the "psh variation" of fiber-wise twisted Kahler-Einstein metrics.