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

Abstract We prove Vojta’s abc conjecture for the projective space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{P}^{n}(\mathbb{C}) , assuming that the entire curves in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{P}^{n}(\mathbb{C}) are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n=2 (see [J. Guo and J. T.-Y. Wang, A complex case of Vojta’s general ABC conjecture and cases of Campana’s orbifold conjecture, Trans. Amer. Math. Soc. 377 (2024), 7, 4961–4991]). 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.