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 > 2 admitting the action of the group G := Z^{n-1} such that every non-trivial element of G is of positive entropy. We show: `X is not rationally connected' ==> `X is G-equivariant birational to the quotient of a complex torus' <==> `K_X + D is pseudo-effective for some G-periodic effective fractional divisor D.' See Main Theorem 2.5. To apply, one uses the above and fact: `the Kodaira dimension of X is at least 0' ==> `X is not uniruled' ==> `X is not rationally connected.' We may generalize the result to the case of solvable G as in Remark 2.7.