Numerically non-special varieties

Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves, i.e. rank-one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question of whether one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank-one coherent subsheaf of the cotangent bundle with numerical dimension one is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results.