Finiteness theorems for complements of large divisors

TL;DR

This paper establishes finiteness of integral points on complements of large divisors in projective varieties over finitely generated fields, extending classical results using advanced Diophantine approximation techniques.

Contribution

It proves a new finiteness theorem for integral points in a broad geometric setting, employing a function field analogue of the Subspace Theorem and a method to transfer results from number fields.

Findings

Finiteness of integral points on complements of large divisors

Extension of classical finiteness results to finitely generated fields

Application of Wang's Subspace Theorem in a new context

Abstract

We prove finiteness results on integral points on complements of large divisors in projective varieties over finitely generated fields of characteristic zero. To do so, we prove a function field analogue of arithmetic finiteness results of Corvaja-Zannier and Levin using Wang's function field Subspace Theorem. We then use a method of Evertse-Gy\H{o}ry for concluding finiteness of integral points over finitely generated fields from known finiteness results over number fields.