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 [18], that is, if X is a normal projective variety over a finite field and D is a pseudo-effective Q-Cartier divisor on X, does it follow that D is Q-effective? We also give affirmative answers on an abelian variety and a projective bundle over a curve.

Algebraic Dynamical Systems and Dirichlet's Unit Theorem on Arithmetic Varieties

“…Even for the algebraic dynamical system as treated in Theorem 0.1, it is a very interesting and challenging problem to find a non-trivial sufficient condition to ensure the Dirichlet property. Further, in [21], we introduce a geometric analogue of the above question. Namely, if D is a pseudo-effective Q-Cartier divisor on a normal projective variety defined over a finite field, can we conclude that D is Q-effective?…”