Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T04:22:32.207Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected] 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 [email protected]
Erwan Rousseau
Affiliation:
Institut Universitaire de France & Aix Marseille Univ., CNRS, Centrale Marseille, I2M, Marseille, [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

REFERENCES

Aihara, Y. and Noguchi, J., Value distribution of meromorphic mappings into compactified locally symmetric spaces, Kodai Math. J. 14 (1991), 320–334; MR 1131916.Google Scholar
Beauville, A., Application aux espaces de modules, in Géométrie des surfaces K3 : modules et périodes, Astérisque, vol. 126 (Société mathématique de France, 1985), 1–193; MR 785231.Google Scholar
Bloch, A., Sur les systèmes de fonctions uniformes satisfaisant à l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension, J. de Math. 5 (1926), 19–66.Google Scholar
Bogomolov, F., Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), 1041–1044.Google Scholar
Brückmann, P. and Rackwitz, H.-G., $T$-symmetrical tensor forms on complete intersections, Math. Ann. 288 (1990), 627–635.Google Scholar
Brunella, M., Courbes entières et feuilletages holomorphes, Enseign. Math. (2) 45 (1999), 195–216.Google Scholar
Campana, F., Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–630.Google Scholar
Campana, F., Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques, Manuscripta Math. 117 (2005), 429–461; MR 2163487.Google Scholar
Campana, F., Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011), 809–934.Google Scholar
Campana, F. and Păun, M., Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861.Google Scholar
Campana, F. and Winkelmann, J., A Brody theorem for orbifolds, Manuscripta Math. 128 (2009), 195–212.Google Scholar
Cartan, H., Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, Ann. Sci. Éc. Norm. Supér. (3) 45 (1928), 255–346; MR 1509288.Google Scholar
Claudon, B., Positivité du cotangent logarithmique et conjecture de Shafarevich–Viehweg, in Séminaire Bourbaki, Astérisque, vol. 390 (Société Mathématique de France, 2017), Exp. No. 1105, 27–63.Google Scholar
Darondeau, L. and Pragacz, P., Universal Gysin formulas for flag bundles, Int. J. Math. 28 (2017), 1750077.Google Scholar
Demailly, J.-P., Vanishing theorems for tensor powers of an ample vector bundle, Invent. Math. 91 (1988), 203–220.Google Scholar
Demailly, J.-P., Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, in Algebraic geometry, Proceedings of Symposia in Pure Mathematics, vol. 62 (American Mathematical Society, Providence, RI, 1997), 285–360; MR 1492539.Google Scholar
Demailly, J.-P., Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q. 7 (2011), 1165–1207; Special Issue: In memory of Eckart Viehweg.Google Scholar
Dethloff, G.-E. and Lu, S. S.-Y., Logarithmic jet bundles and applications, Osaka J. Math. 38 (2001), 185–237.Google Scholar
Diverio, S., Differential equations on complex projective hypersurfaces of low dimension, Compos. Math. 144 (2008), 920–932.Google Scholar
Diverio, S., Existence of global invariant jet differentials on projective hypersurfaces of high degree, Math. Ann. 344 (2009), 293–315.10.1007/s00208-008-0306-4CrossRefGoogle Scholar
El Goul, J., Logarithmic jets and hyperbolicity, Osaka J. Math. 40 (2003), 469–491; MR 1988702.Google Scholar
Green, M. and Griffiths, P., Two applications of algebraic geometry to entire holomorphic mappings, in The Chern symposium 1979 (Springer, New York, NY, 1980), 41–74.Google Scholar
Guenancia, H. and Păun, M., Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors, J. Differential Geom. 103 (2016), 15–57.10.4310/jdg/1460463562CrossRefGoogle Scholar
Huynh, D. T., Vu, D.-V. and Xie, S.-Y., Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree, Ann. Inst. Fourier (Grenoble) 69 (2019), 653–671.Google Scholar
Kobayashi, S., Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318 (Springer, Berlin, 1998).Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge [Results in Mathematics and Related Areas. 3rd Series], vol. 48 (Springer, Berlin, 2004).Google Scholar
Manivel, L., Un théorème d'annulation “à la Kawamata-Viehweg”, Manuscripta Math. 83 (1994), 387–404.Google Scholar
McQuillan, M., Diophantine approximations and foliations, Publ. Math. Inst. Hautes Études Sci. 87 (1998), 121–174; MR 1659270.Google Scholar
Merker, J., Algebraic differential equations for entire holomorphic curves in projective hypersurfaces of general type: optimal lower degree bound, in Geometry and analysis on manifolds, Progress in Mathematics, vol. 308 (Birkhäuser/Springer, Cham, 2015), 41–142.Google Scholar
Noguchi, J., On the value distribution of meromorphic mappings of covering spaces over $\mathbb {C}^{m}$ into algebraic varieties, J. Math. Soc. Japan 37 (1985), 295–313.Google Scholar
Noguchi, J., Logarithmic jet spaces and extensions of de Franchis’ theorem, in Contributions to several complex variables, Aspects of Mathematics E9 (Friedr. Vieweg, Braunschweig, 1986), 227–249.CrossRefGoogle Scholar
Păun, M. and Sibony, N., Value distribution theory for parabolic riemann surfaces, Preprint (2014), arXiv:1403.6596.Google Scholar
Rousseau, E., Hyperbolicity of geometric orbifolds, Trans. Amer. Math. Soc. 362 (2010), 3799–3826.Google Scholar
Rousseau, E., Degeneracy of holomorphic maps via orbifolds, Bull. Soc. Math. France 140 (2012), 459–484.CrossRefGoogle Scholar
Ru, M., Nevanlinna theory and its relation to Diophantine approximation (World Scientific, River Edge, NJ, 2001).10.1142/4508CrossRefGoogle Scholar
Ru, M., Holomorphic curves into algebraic varieties, Ann. Math. (2) 169 (2009), 255–267; MR 2480605.Google Scholar
Siu, Y.-T. and Yeung, S.-K., Addendum to: “Defects for ample divisors of Abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees” [Amer. J. Math 119 (1997), 1139–1172; MR 1473072 (98h:32044)], Amer. J. Math. 125 (2003), 441–448.Google Scholar
Vojta, P., Diophantine approximation and Nevanlinna theory, in Arithmetic geometry, Lecture Notes in Mathematics, vol. 2009 (Springer, Berlin, 2011), 111–224; MR 2757629.CrossRefGoogle Scholar
Yamanoi, K., Algebro-geometric version of Nevanlinna's lemma on logarithmic derivative and applications, Nagoya Math. J. 173 (2004), 23–63.10.1017/S0027763000008710CrossRefGoogle Scholar
Yamanoi, K., Holomorphic curves in Abelian varieties and intersections with higher codimensional subvarieties, Forum Math. 16 (2004), 749–788; MR 2096686.CrossRefGoogle Scholar
Yamanoi, K., Kobayashi hyperbolicity and higher-dimensional Nevanlinna theory, in Geometry and analysis on manifolds, Progress in Mathematics, vol. 308 (Birkhäuser/Springer, Cham, 2015), 209–273; MR 3331401.CrossRefGoogle Scholar