Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Let f: X → A be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves f_∗ωX^⊗m become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when m ≥ 2, analogous to that obtained by Chen-Jiang when m = 1. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.