Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary

TL;DR

This paper proves Vojta's abc conjecture for entire curves in projective space and toric varieties under high ramification conditions, extending previous results and addressing Campana's orbifold conjecture.

Contribution

It extends Vojta's abc conjecture to higher-dimensional projective spaces and toric varieties with high ramification, and establishes a version of Campana's orbifold conjecture.

Findings

Vojta's abc conjecture proven for ${f P}^n({f C})$ with high ramification

Results extended to projective toric varieties

Established a version of Campana's orbifold conjecture

Abstract

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case $n=2$ (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.