Pushforwards of klt pairs under morphisms to abelian varieties

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