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

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