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Let $X$ be a normal projective variety of dimension $n$ and $G$ an abeliangroup of automorphisms such that all elements of $G\setminus \{\mathrm{id}\}$are of positive entropy. Dinh and Sibony showed that $G$ is actually freeabelian of rank $\le n - 1$. The maximal rank case has been well understood byDe-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\mathrm{rank}G = n - 2$.

Free abelian group actions on normal projective varieties: sub-maximal dynamical rank case