  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.
