Parshin's method and the geometric Bombieri-Lang conjecture

TL;DR

This paper surveys Parshin's proof of the geometric Bombieri-Lang conjecture and demonstrates its application in providing an alternative proof for specific cases resolved by Xie-Yuan.

Contribution

It explains Parshin's method and shows its use in deriving an alternative proof for the geometric Bombieri-Lang conjecture in certain varieties.

Findings

Parshin's proof can be applied to varieties with finite morphisms to traceless abelian varieties.

An alternative proof of the geometric Bombieri-Lang conjecture is provided for specific cases.

The survey clarifies the connection between Parshin's method and recent results by Xie-Yuan.

Abstract

In this short survey, we explain Parshin's proof of the geometric Bombieri-Lang conjecture, and show that it can be used to give an alternative proof of Xie-Yuan's recent resolution of the geometric Bombieri-Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety.