Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{ \mathbb Q}m(K_X+\Delta)$. We prove that the sheaf $f_*\mathcal{ O}_X(D)$ becomes globally generated after pullback by an isogeny and has the Chen-Jiang decomposition, along with some related results. These are applied to some effective results for $\mathcal{ O}_X(D)$ when $X$ is irregular.
Categories: math.AG