Geometric Bogomolov conjecture in arbitrary characteristics

TL;DR

This paper proves the full geometric Bogomolov conjecture across arbitrary characteristics by reducing the problem to a transcendence degree 1 case and applying intersection theory, building on Yamaki's reduction and classical conjectures.

Contribution

It provides a complete proof of the geometric Bogomolov conjecture in all characteristics, extending previous partial results and utilizing advanced algebraic geometry techniques.

Findings

Confirmed the conjecture in arbitrary characteristics

Reduced the problem to a transcendence degree 1 case

Applied intersection theory in the proof

Abstract

The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.