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 ≥ 3 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 Gequivariant birational to the quotient of a complex torus’ ⇐⇒ ‘KX + D is pseudo-effective for some G-periodic effective fractional divisor D’. To apply, one uses the above and the fact: ‘the Kodaira dimension κ(X) ≥ 0’ ⇒ ‘X is not uniruled’ ⇒ ‘X is not rationally connected’. We may generalize the result to the case of solvable G.