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

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 $\mathbb G_m<sup>n$ 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 $\mathbb P<sup>n$ 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.