Campana’s characterization of special varieties by integral points

Establish that a variety X over a number field K is Campana-special if and only if the set of integral points on X is potentially dense.

Background

Campana introduced the notion of special varieties to capture certain geometric behaviors. He conjectured an arithmetic characterization equating specialness with the potential density of integral points.

This forms a central open problem connecting the geometry of special varieties with Diophantine properties.

References

Conjecture [Campana]\label{conj:special} Let $X$ be a variety over a number field $K$. Then $X$ is special if and only if the set of integral points on $X$ is potentially dense.

— Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell  (2603.28745 - Bartsch et al., 30 Mar 2026) in Introduction, Special varieties (Conjecture \ref{conj:special})