Vojta’s weak conjecture: height–proximity inequality

Establish the Vojta height–proximity inequality for smooth projective varieties over a number field: given a smooth projective variety X over K, an effective divisor D ⊂ X, and a finite set of places S of K, prove that for every ε > 0 there exist a proper closed subset Z ⊂ X and a constant C_ε such that for all points P ∈ X(\overline{K}) \ Z one has h_{K_X + D}(P) ≤ (1 + ε) m_S(P, D) + C_ε, where m_S(P, D) denotes the S-proximity function to D and h_{K_X + D} is the height associated to K_X + D.

Background

The paper situates its results within Vojta’s conjectural framework linking Diophantine inequalities to Nevanlinna theory. In Section 6.1, the authors explicitly state a weak form of Vojta’s conjecture that predicts a global inequality comparing the height associated to K_X + D with the S-proximity to D, outside a proper closed subset.

They note that their main theorem provides partial progress under positivity assumptions (e.g., when K_X + D is big), but the general conjecture without such assumptions remains unresolved. The stated conjecture serves as the central open problem guiding the framework developed in the paper.