Toward a geometric analogue of Dirichlet’s unit theorem

Abstract

In this article, we propose a geometric analogue of Dirichlet’s unit theorem on arithmetic varieties; that is, if $X$ is a normal projective variety over a finite field and $D$ is a pseudo-effective $\mathbb{Q}$-Cartier divisor on $X$, does it follow that $D$ is $\mathbb{Q}$-effective? We also give affirmative answers on an abelian variety and a projective bundle over a curve.