Intersection pairing for arithmetic cycles with degenerate Green currents

TL;DR

This paper extends the arithmetic Chow group to include degenerate Green currents, ensuring the Hodge index theorem applies, and proves an arithmetic analogue of Bogomolov's instability theorem for rank 2 bundles.

Contribution

It introduces a new extension of the arithmetic Chow group and establishes an arithmetic Bogomolov instability theorem for vector bundles.

Findings

Hodge index theorem holds in the extended setting

Arithmetic analogue of Bogomolov's instability theorem proved

Extension accommodates degenerate Green currents

Abstract

In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.