Campana’s generalized Lang conjecture for C-pairs

Prove that if (X,Δ) is a smooth proper C-pair of general type over a number field K with dim X ≥ 1, then for any finite set S of finite places and any smooth proper model over O_{K,S}, the set of O_{K,S}-integral points on (X,Δ) is not dense in X.

Background

This conjecture extends Lang’s conjecture to the orbifold (C-pair) setting, predicting non-density of integral points whenever the orbifold is of general type.

The paper shows that a strong form of the heuristic “arithmetically special implies special” would imply this conjecture, highlighting its centrality in the arithmetic of C-pairs.

References

Conjecture [Campana's generalized Lang conjecture]\label{conj:gen_lang} Let $(X,\sum_{i=1}r (1-\frac{1}{m_i})D_i)$ be a smooth proper C-pair of general type over a number field $K$. Let $S$ be a finite set of finite places of $K$ and let $\mathcal{X}\to \Spec \mathcal{O}{K,S}$ be a smooth proper model of $X$ over $\mathcal{O}{K,S}$ and let $\mathcal{D}i$ be the closure of $D_i$ in $\mathcal{X}$. Write $\Delta =\sum{i=1}r\left(1-\frac{1}{m_i}\right)\mathcal{D}_i$. If $\dim X \geq 1$, then $(\mathcal{X},\Delta)(\mathcal{O}_{K,S})$ is not dense in $X$.

— Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell  (2603.28745 - Bartsch et al., 30 Mar 2026) in Section “Campana’s generalized Lang conjecture” (Conjecture \ref{conj:gen_lang})