The arithmetic Hodge index theorem for adelic line bundles I

TL;DR

This paper establishes an arithmetic Hodge index theorem for adelic line bundles on projective varieties over number fields, extending previous results and deriving implications for non-archimedean Calabi--Yau theorem and dynamical systems.

Contribution

It introduces a new arithmetic Hodge index theorem for adelic line bundles, generalizing prior theorems and connecting to dynamical systems and Calabi--Yau properties.

Findings

Proves an arithmetic Hodge index theorem for adelic line bundles.

Derives the uniqueness part of the non-archimedean Calabi--Yau theorem.

Establishes a rigidity property for preperiodic points in algebraic dynamical systems.

Abstract

This is the first paper of a series. We prove an arithmetic Hodge index theorem for adelic line bundles on projective varieties over number fields. It extends the arithmetic Hodge index theorem of Faltings, Hriljac and Moriwaki on arithmetic varieties. As consequences, we obtain the uniqueness part of the non-archimedean Calabi--Yau theorem, and a rigidity property of the sets of preperiodic points of polarizable algebraic dynamical systems over number fields.