Positivity properties of pushforwards of pluricanonical bundles under morphisms to abelian varieties

Let f : X --> A a morphism from a smooth projective complex variety to an abelian variety. We will show that the direct images of pluricanonical bundles of X under f are globally generated up to a pull-back of an isogeny of A of finite degree. As an application, we will show that such sheaves admit direct-sum decompositions whose summands satisfy a certain