Periodic subvarieties of a projective variety under the action of a maximal rank Abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an ndimensional normal projective variety X under the action of an abelian group G of maximal rank n - 1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.