  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 if 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 1 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.
