Free abelian group actions on projective fibered varieties

Let $X$ be an $n$-dimensional normal projective variety of Kodaira dimension $\kappa$ and $G$ an abelian subgroup of automorphisms such that all elements of $G\mathrm{\backslash }\{\mathrm{i}\mathrm{d}\}$ are of positive entropy. Dinh and Sibony showed that $G$ is actually free abelian of rank $\le n-\kappa -1$. In this paper, we show that after replacing $G$ by a finite-index subgroup, there exists a G-Equivariant Minimal Model Program for the pair $(X,G)$ when $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(G)=n-\kappa -1$ and $\kappa >0$, and the very general fiber of the final model is a klt Calabi-Yau surface or a $Q$-abelian variety provided that the termination of flips holds in this setting.