The theta invariants and the volume function on arithmetic varieties

TL;DR

This paper introduces a new invariant for hermitian line bundles on arithmetic varieties, using it to analyze the volume function's variation and establishing a generalized Hodge index theorem for arithmetic toric varieties.

Contribution

It proposes a novel arithmetic invariant and applies it to extend the Hodge index theorem to arithmetic toric varieties.

Findings

New invariant for hermitian line bundles

Generalized Hodge index theorem on arithmetic toric varieties

Insights into volume function variation with metrics

Abstract

We introduce a new arithmetic invariant for hermitian line bundles on an arithmetic variety. We use this invariant to measure the variation of the volume function with respect to the metric. The main result of this paper is a generalized Hodge index theorem on arithmetic toric varieties.