n-DIMENSIONAL PROJECTIVE VARIETIES WITH THE ACTION OF AN ABELIAN GROUP OF RANK n-1

Let X be a normal projective variety of dimension n >= 3 admitting the action of the group G := Z(circle plus n-1) such that every non-trivial element of G is of positive entropy. We show: 'X is not rationally connected' double right arrow 'X is G-equivariant birational to the quotient of a complex torus' double left arrow double right arrow '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) >= 0' double right arrow 'X is not uniruled' double right arrow 'X is not rationally connected'. We may generalize the result to the case of solvable G.