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

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.

💡 Research Summary

The paper establishes Vojta’s abc conjecture for entire holomorphic curves in complex projective space ℙⁿ(ℂ) under the hypothesis that the curves are highly ramified over the coordinate hyperplanes. The authors assume a homogeneous polynomial G∈ℂ