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arXiv · 2106.15881

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

Ji Guo·Khoa D. Nguyen·Chia-Liang Sun·Julie Tzu-Yueh Wang
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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.

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Ji Guo, Khoa D. Nguyen, Chia-Liang Sun, Julie Tzu-Yueh Wang. 2021-06-30. Vojta's abc Conjecture for algebraic tori and applications over function fields. https://arxiv.org/abs/2106.15881

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