# Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture

## TL;DR

This paper introduces Arakelov geometry and its applications to heights, equidistribution, and the proof of the Bogomolov conjecture, emphasizing the interplay between classical and adelic frameworks.

## Contribution

It reviews key concepts in Arakelov geometry, explains height properties, and discusses the proof of the Bogomolov conjecture using equidistribution techniques.

## Key findings

- Explanation of arithmetic intersection numbers
- Discussion of height properties in Arakelov geometry
- Outline of Ullmo-Zhang's proof of the Bogomolov conjecture

## Abstract

This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.