Vojta's abc Conjecture for algebraic tori and applications over function fields

TL;DR

This paper proves a generalized abc conjecture for algebraic tori over function fields, establishes related results for toric varieties, and explores applications to the Lang-Vojta and Green-Griffith-Lang conjectures, providing explicit exceptional sets.

Contribution

It extends Vojta's abc conjecture to algebraic tori and toric varieties over function fields, with effective exceptional sets and applications to major conjectures.

Findings

Proved Vojta's abc conjecture for algebraic tori over function fields.

Established a version of the conjecture for toric varieties.

Derived explicit exceptional sets for related complex cases.

Abstract

We prove Vojta's generalized abc conjecture for algebraic tori over function fields with exceptional sets that can be determined effectively. Additionally, we establish a version of the conjecture for toric varieties. As an application, we investigate the Lang-Vojta Conjecture for varieties of log general type that are ramified covers of $\mathbb G_m^n$ over function fields. In particular, we consider the case of $ \mathbb P^n\setminus D$, where $D$ is an algebraic curve over a function field in $\mathbb P^n$ with $n+1$ irreducible components and $\deg D\ge n+2$. Our methods also apply to the complex situation, enabling us to find explicit exceptional sets for the corresponding case of Vojta's general abc conjecture (complex version) and the Green-Griffith-Lang conjecture.