No CrossRef data available.
Article contents
Numerically non-special varieties
Published online by Cambridge University Press: 22 August 2022
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.
MSC classification
Primary:
32J27: Compact Kähler manifolds
- Type
- Research Article
- Information
- Copyright
- © 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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.
References
Baily, W. L., On the imbedding of 
$V$-manifolds in projective space, Amer. J. Math. 79 (1957), 403–430.CrossRefGoogle Scholar
$V$-manifolds in projective space, Amer. J. Math. 79 (1957), 403–430.CrossRefGoogle ScholarBoucksom, S., On the volume of a line bundle, Internat. J. Math. 13 (2002), 1043–1063.CrossRefGoogle Scholar
Campana, F., Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–630.CrossRefGoogle Scholar
Campana, F., Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques, Manuscripta Math. 117 (2005), 429–461.CrossRefGoogle Scholar
Campana, F., Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011), 809–934.CrossRefGoogle Scholar
Campana, F., Arithmetic aspects of orbifold pairs, in Arithmetic geometry of logarithmic pairs and hyperbolicity of moduli spaces—hyperbolicity in Montréal, CRM Short Courses (Springer, Cham, 2020), 69–133.Google Scholar
Campana, F., Claudon, B., and Eyssidieux, P., Représentations linéaires des groupes kählériens: factorisations et conjecture de Shafarevich linéaire, Compos. Math. 151 (2015), 351–376.CrossRefGoogle Scholar
Cadorel, B., Diverio, S. and Guenancia, H., On subvarieties of singular quotients of bounded domains, J. Lond. Math. Soc. (2), to appear. Preprint (2020), arXiv:1905.04212.Google Scholar
Corlette, K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361–382.CrossRefGoogle Scholar
$G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361–382.CrossRefGoogle ScholarDebarre, O., Higher-dimensional algebraic geometry, Universitext (Springer, New York, 2001).CrossRefGoogle Scholar
Demailly, J.-P., Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), 361–409.Google Scholar
Demailly, J.-P., Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, in Algebraic geometry—Santa Cruz 1995, Proceedings of Symposia in Pure Mathematics, vol. 62 (American Mathematical Society, Providence, RI, 1997), 285–360.CrossRefGoogle Scholar
Demailly, J.-P., On the Frobenius integrability of certain holomorphic p-forms, Complex Geometry (Göttingen, 2000) (Springer, Berlin, 2002), 93–98.Google Scholar
El Kacimi-Alaoui, A., Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compos. Math. 73 (1990), 57–106.Google Scholar
Granath, H., On quaternionic Shimura surfaces, PhD thesis, Chalmers Tekniska Hogskola (Sweden) (ProQuest LLC, Ann Arbor, MI, 2002).Google Scholar
Hironaka, H., Bimeromorphic smoothing of a complex-analytic space, Acta Math. Vietnam. 2 (1977), 103–168.Google Scholar
Javanpeykar, A. and Rousseau, E., Albanese maps and fundamental groups of varieties with many rational points over function fields, Int. Math. Res. Not. IMRN (2022), to appearCrossRefGoogle Scholar
Labourie, F., Existence d'applications harmoniques tordues à valeurs dans les variétés à courbure négative, Proc. Amer. Math. Soc. 111 (1991), 877–882.Google Scholar
Lo Bianco, F., Pereira, J., Rousseau, E. and Touzet, F., Rational endomorphisms of codimension one holomorphic foliations, Preprint (2020), arXiv:2007.12541.Google Scholar
Mok, N., The generalized theorem of Castelnuovo-de Franchis for unitary representations, Geometry from the Pacific Rim (Singapore, 1994) (de Gruyter, Berlin, 1997), 261–284.Google Scholar
Mok, N., Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups, Ann. Inst. Fourier (Grenoble) 50 (2000), 633–675.CrossRefGoogle Scholar
Molino, P., Riemannian foliations, Progress in Mathematics, vol. 73 (Birkhäuser, Boston, MA, 1988). Translated from the French by Grant Cairns, with appendices by Cairns, Y. Carrière, É. Ghys, E. Salem, and V. Sergiescu.CrossRefGoogle Scholar
Nevanlinna, R., Analytic functions, Grundlehren der mathematischen Wissenschaften, Band 162 (Springer, New York, Berlin, 1970). Translated from the second German edition by Phillip Emig.CrossRefGoogle Scholar
Rousseau, E. and Touzet, F., Curves in Hilbert modular varieties, Asian J. Math. 22 (2018), 673–690.CrossRefGoogle Scholar
Shimizu, H., On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33–71.CrossRefGoogle Scholar
Touzet, F., Structure des feuilletages kähleriens en courbure semi-négative, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), 865–886.CrossRefGoogle Scholar
Touzet, F., Uniformisation de l'espace des feuilles de certains feuilletages de codimension un, Bull. Braz. Math. Soc. (N.S.) 44 (2013), 351–391.CrossRefGoogle Scholar
Touzet, F., On the structure of codimension 1 foliations with pseudoeffective conormal bundle, in Foliation theory in algebraic geometry, Simons Symposia (Springer, Cham, 2016), 157–216.CrossRefGoogle Scholar
Vâjâitu, V., Kählerianity of $q$-Stein spaces, Arch. Math. (Basel) 66 (1996), 250–257.CrossRefGoogle Scholar
$q$-Stein spaces, Arch. Math. (Basel) 66 (1996), 250–257.CrossRefGoogle ScholarZuo, K., Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of $\pi _1$ of compact Kähler manifolds, J. Reine Angew. Math. 472 (1996), 139–156.Google Scholar
$\pi _1$ of compact Kähler manifolds, J. Reine Angew. Math. 472 (1996), 139–156.Google Scholar