$n$-dimensional projective varieties with the action of an abelian group of rank $n-1$

Abstract:

Let $X$ be a normal projective variety of dimension $n \ge 3$ admitting the action of the group $G := \mathbb {Z}^{\oplus n-1}$ such that every non-trivial element of $G$ is of positive entropy. We show: ‘$X$ is not rationally connected’ $\Rightarrow$ ‘$X$ is $G$-equivariant birational to the quotient of a complex torus’ $\Leftarrow \Rightarrow$ ‘$K_X + D$ is pseudo-effective for some $G$-periodic effective fractional divisor $D$’. To apply, one uses the above and the fact: ‘the Kodaira dimension $\kappa (X) \ge 0$’ $\Rightarrow$ ‘$X$ is not uniruled’ $\Rightarrow$ ‘$X$ is not rationally connected’. We may generalize the result to the case of solvable $G$.