Nous étudions l’hyperbolicité des quotients singuliers de domaines symétriques bornés. Nous donnons des critères effectifs assurant que de tels quotients vérifient les conjectures de Green-Griffiths-Lang, à la fois dans le cadre analytique et algébrique. Comme application, nous établissons que les variétés modulaires de Hilbert, à part quelques exceptions possibles, satisfont les conjectures attendues.
We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Lang’s conjectures in both analytic and algebraic settings. As an application, we show that Hilbert modular varieties, except for a few possible exceptions, satisfy all expected conjectures.
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DOI : 10.5802/jep.85
Keywords: Green-Griffiths-Lang’s conjectures, bounded symmetric domains, quotient singularities, Hilbert modular varieties
Mot clés : Conjectures de Green-Griffiths-Lang, domaines symétriques bornés, singularités quotients, variétés modulaires de Hilbert
@article{JEP_2019__6__1_0, author = {Cadorel, Beno{\^\i}t and Rousseau, Erwan and Taji, Behrouz}, title = {Hyperbolicity of singular spaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1--18}, publisher = {Ecole polytechnique}, volume = {6}, year = {2019}, doi = {10.5802/jep.85}, zbl = {07003359}, mrnumber = {3882579}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.85/} }
TY - JOUR AU - Cadorel, Benoît AU - Rousseau, Erwan AU - Taji, Behrouz TI - Hyperbolicity of singular spaces JO - Journal de l’École polytechnique — Mathématiques PY - 2019 SP - 1 EP - 18 VL - 6 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.85/ DO - 10.5802/jep.85 LA - en ID - JEP_2019__6__1_0 ER -
Cadorel, Benoît; Rousseau, Erwan; Taji, Behrouz. Hyperbolicity of singular spaces. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 1-18. doi : 10.5802/jep.85. http://www.numdam.org/articles/10.5802/jep.85/
[AMRT75] Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, IV, Math. Sci. Press, Brookline, MA, 1975 | Zbl
[BC18] Hyperbolicity of varieties supporting a variation of Hodge structure, Internat. Math. Res. Notices (2018) (article no. rny054) | DOI
[BD18] A note on Lang’s conjecture for quotients of bounded domains (2018) (arXiv:1809.02398)
[Bor69] Introduction aux groupes arithmétiques, Actualités scientifiques et industrielles, Hermann, Paris, 1969 | Zbl
[Bru16] A strong hyperbolicity property of locally symmetric varieties (2016) (arXiv:1606.03972)
[Cad16] Symmetric differentials on complex hyperbolic manifolds with cusps (2016) (arXiv:1606.05470)
[Cam04] Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 3, pp. 499-630 | DOI | MR | Zbl
[CKT16] Generic positivity and applications to hyperbolicity of moduli spaces (2016) (arXiv:1610.09832)
[CP15] Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble), Volume 65 (2015) no. 2, pp. 835-861 | DOI | MR | Zbl
[Dem97] Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, American Mathematical Society, Providence, RI, 1997, pp. 285-360 | DOI | MR | Zbl
[GG80] Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York, 1980, pp. 41-74 | DOI | Zbl
[GKK10] Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties, Compositio Math., Volume 146 (2010), pp. 193-219 (Extended version: arXiv:0808.3647) | DOI | MR | Zbl
[GP16] Conic singularities metrics with prescribed Ricci curvature: General cone angles along normal crossing divisors, J. Differential Geom., Volume 103 (2016) no. 1, pp. 15-57 | DOI | MR | Zbl
[GRVR18] On Lang’s conjecture for some product-quotient surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 18 (2018) no. 4, pp. 1483-1501 | MR | Zbl
[GT16] Orbifold stability and Miyaoka-Yau inequality for minimal pairs (2016) (arXiv:1611.05981)
[Gue18] Quasi-projective manifolds with negative holomorphic sectional curvature (2018) (arXiv:1808.01854)
[JK11] Families over special base manifolds and a conjecture of Campana, Math. Z., Volume 269 (2011) no. 3-4, pp. 847-878 | DOI | MR | Zbl
[Keu08] Quotients of fake projective planes, Geom. Topol., Volume 12 (2008) no. 4, pp. 2497-2515 | DOI | MR | Zbl
[KM08] Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, Cambridge University Press, 2008
[Kol07] Lectures on resolution of singularities, Annals of Math. Studies, 166, Princeton University Press, Princeton, NJ, 2007 | MR | Zbl
[Lan86] Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.), Volume 14 (1986) no. 2, pp. 159-205 | DOI | MR | Zbl
[Laz04] Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergeb. Math. Grenzgeb. (3), 48, Springer-Verlag, Berlin, 2004 | Zbl
[Mum77] Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272 | DOI | MR | Zbl
[Nad89] The nonexistence of certain level structures on abelian varieties over complex function fields, Ann. of Math. (2), Volume 129 (1989) no. 1, pp. 161-178 | DOI | MR | Zbl
[Pre68] Die elliptischen Fixpunkte der Hilbertschen Modulgruppen, Math. Ann., Volume 177 (1968), pp. 181-209 | DOI | MR | Zbl
[Rou16] Hyperbolicity, automorphic forms and Siegel modular varieties, Ann. Sci. École Norm. Sup. (4), Volume 49 (2016) no. 1, pp. 249-255 | DOI | MR | Zbl
[RT18] Curves in Hilbert modular varieties, Asian J. Math., Volume 22 (2018) no. 4, pp. 673-690 | DOI | MR | Zbl
[Taj16] The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base, Compositio Math., Volume 152 (2016) no. 7, pp. 1421-1434 | DOI | MR | Zbl
[Tsu85] On the Kodaira dimensions of Hilbert modular varieties, Invent. Math., Volume 80 (1985) no. 2, pp. 269-281 | DOI | MR | Zbl
[Tsu86] Multitensors of differential forms on the Hilbert modular variety and on its subvarieties, Math. Ann., Volume 274 (1986) no. 4, pp. 659-670 | DOI | MR | Zbl
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