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Orbifold hyperbolicity

Part of: Complex manifolds Holomorphic mappings and correspondences Holomorphic fiber spaces

Published online by Cambridge University Press:  08 October 2020

Frédéric Campana ,
Lionel Darondeau  and
Erwan Rousseau
Frédéric Campana
Affiliation:
Institut de Mathématiques Élie Cartan, Université de Lorraine, B.P. 70239, 54506Vandœuvre-lés-Nancy Cedex, Francefrederic.campana@univ-lorraine.fr KIAS, 85 Hoegiro, Dongdaemungu, Seoul130-722, South Korea
Lionel Darondeau
Affiliation:
KU Leuven, Departement Wiskunde, Celestijnenlaan 200B, 3001Heverlee, BelgiëCurrent address: IMAG, Univ. Montpellier, CNRS, Montpellier, France lionel.darondeau@normalesup.org
Erwan Rousseau
Affiliation:
Institut Universitaire de France & Aix Marseille Univ., CNRS, Centrale Marseille, I2M, Marseille, Franceerwan.rousseau@univ-amu.fr
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Abstract

We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green–Griffiths–Lang conjecture. This contrasts with an important result of Demailly (Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q. 7 (2011), 1165–1207) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.

Keywords

Green–Griffiths–Lang's conjecturesorbifold pairsvanishing theoremsentire curvesKobayashi hyperbolicity

MSC classification

Primary: 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds 32H30: Value distribution theory in higher dimensions 32L20: Vanishing theorems
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 8 , August 2020 , pp. 1664 - 1698
Copyright
Copyright © The Author(s) 2020

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Footnotes

This work has been carried out in the framework of Archimède Labex (ANR-11-LABX-0033) and of the AMIDEX project (ANR-11-IDEX-0001-02), funded by the ‘Investissements d'Avenir’ French Government program managed by the French National Research Agency (ANR). Erwan Rousseau was partially supported by the ANR project ‘FOLIAGE’, ANR-16-CE40-0008. Lionel Darondeau is a postdoctoral fellow of The Research Foundation – Flanders (FWO). L.D and E.R. thank the KIAS where part of this work was done.

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