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Numerically non-special varieties

Part of: Compact analytic spaces Global differential geometry

Published online by Cambridge University Press:  22 August 2022

Jorge Vitório Pereira ,
Erwan Rousseau  and
Frédéric Touzet
Jorge Vitório Pereira
Affiliation:
IMPA, Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, Brasil [email protected]
Erwan Rousseau
Affiliation:
Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France [email protected]
Frédéric Touzet
Affiliation:
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France [email protected]
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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 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.

Keywords

special varietiesBogomolov sheavesentire curves

MSC classification

Primary: 32J27: Compact Kähler manifolds
Secondary: 53C12: Foliations (differential geometric aspects)
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 6 , June 2022 , pp. 1428 - 1447
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

This work was supported by the ANR project ‘FOLIAGE’, ANR-16-CE40-0008 and CAPES-COFECUB Ma 932/19 project. The first author was supported by Cnpq and FAPERJ.

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