  We explore the relationship between fibrations arising naturally from a
surjective morphism to an abelian variety. These fibrations encode geometric
information about the morphism. Our study focuses on the interplay of these
fibrations and presents several applications. Then we propose a generalization
of the Iitaka conjecture which predicts an equality of Kodaira dimension of
fibrations, and prove it when the base is a smooth projective variety of
maximal Albanese dimension.
