Numerically nonspecial varieties

TL;DR

This paper explores the concept of special varieties, examining whether numerical dimension can replace Kodaira dimension in their characterization, and provides partial answers along with analytic and arithmetic perspectives.

Contribution

It shows that projective manifolds with certain cotangent bundle subsheaves are not special and extends the characterization to analytic and arithmetic contexts.

Findings

Manifolds with rank one cotangent subsheaves of numerical dimension 1 are not special.

Established an analytic criterion involving non-existence of Zariski dense entire curves.

Provided an arithmetic perspective with non-potential density in function fields.

Abstract

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.