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

Abstract

The effectivity of pseudo-effective |$\mathbb R$|-divisor is an important and difficult problem in algebraic geometry. In the Arakelov geometry framework, this problem can be considered as a generalization of Dirichlet's unit theorem for number fields. In this article, we propose obstructions to the Dirichlet property by two approaches, that is, the denseness of nonpositive points and functionals on adelic |$\mathbb R$|-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic |${\mathbb {R}}$|-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.