On the conjectures of Vojta and Campana over function fields with explicit exceptional sets

TL;DR

This paper advances the understanding of Vojta's and Campana's conjectures over function fields by proving new cases with explicit exceptional sets, using local analysis of omega-integral varieties.

Contribution

It provides new proven cases of Vojta's conjecture for surfaces and a general explicit result towards Campana's conjecture over complex function fields.

Findings

Proved new cases of Vojta's conjecture with explicit exceptional sets.

Established a general explicit result towards Campana's conjecture.

Utilized local study of omega-integral varieties for proofs.

Abstract

We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result towards Campana's conjecture over complex function fields of curves. Our methods rely on a local study of $\omega$-integral varieties.