[Hyperbolicité des espaces singuliers]
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.
Accepté le :
Publié le :
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
Benoît Cadorel 1 ; Erwan Rousseau 2 ; Behrouz Taji 3
CC-BY 4.0
@article{JEP_2019__6__1_0,
author = {Beno{\^\i}t Cadorel and Erwan Rousseau and Behrouz Taji},
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},
mrnumber = {3882579},
zbl = {07003359},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.85/}
}
TY - JOUR AU - Benoît Cadorel AU - Erwan Rousseau AU - Behrouz Taji TI - Hyperbolicity of singular spaces JO - Journal de l’École polytechnique — Mathématiques PY - 2019 SP - 1 EP - 18 VL - 6 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.85/ DO - 10.5802/jep.85 LA - en ID - JEP_2019__6__1_0 ER -
%0 Journal Article %A Benoît Cadorel %A Erwan Rousseau %A Behrouz Taji %T Hyperbolicity of singular spaces %J Journal de l’École polytechnique — Mathématiques %D 2019 %P 1-18 %V 6 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.85/ %R 10.5802/jep.85 %G en %F JEP_2019__6__1_0
Benoît Cadorel; Erwan Rousseau; Behrouz Taji. Hyperbolicity of singular spaces. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 1-18. doi : 10.5802/jep.85. https://jep.centre-mersenne.org/articles/10.5802/jep.85/
[AMRT75] - Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, vol. 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) 54 (2004) no. 3, p. 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) 65 (2015) no. 2, p. 835-861 | DOI | MR | Zbl
[Dem97] - “Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997, p. 285-360 | DOI | MR | Zbl
[GG80] - “Two applications of algebraic geometry to entire holomorphic mappings”, in The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York, 1980, p. 41-74 | DOI | Zbl
[GKK10] - “Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties”, Compositio Math. 146 (2010), p. 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. 103 (2016) no. 1, p. 15-57 | DOI | MR | Zbl
[GRVR18] - “On Lang’s conjecture for some product-quotient surfaces”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 18 (2018) no. 4, p. 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. 269 (2011) no. 3-4, p. 847-878 | DOI | MR | Zbl
[Keu08] - “Quotients of fake projective planes”, Geom. Topol. 12 (2008) no. 4, p. 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, vol. 166, Princeton University Press, Princeton, NJ, 2007 | MR | Zbl
[Lan86] - “Hyperbolic and Diophantine analysis”, Bull. Amer. Math. Soc. (N.S.) 14 (1986) no. 2, p. 159-205 | DOI | MR | Zbl
[Laz04] - Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | Zbl
[Mum77] - “Hirzebruch’s proportionality theorem in the noncompact case”, Invent. Math. 42 (1977), p. 239-272 | DOI | MR | Zbl
[Nad89] - “The nonexistence of certain level structures on abelian varieties over complex function fields”, Ann. of Math. (2) 129 (1989) no. 1, p. 161-178 | DOI | MR | Zbl
[Pre68] - “Die elliptischen Fixpunkte der Hilbertschen Modulgruppen”, Math. Ann. 177 (1968), p. 181-209 | DOI | MR | Zbl
[Rou16] - “Hyperbolicity, automorphic forms and Siegel modular varieties”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 1, p. 249-255 | DOI | MR | Zbl
[RT18] - “Curves in Hilbert modular varieties”, Asian J. Math. 22 (2018) no. 4, p. 673-690 | DOI | MR | Zbl
[Taj16] - “The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base”, Compositio Math. 152 (2016) no. 7, p. 1421-1434 | DOI | MR | Zbl
[Tsu85] - “On the Kodaira dimensions of Hilbert modular varieties”, Invent. Math. 80 (1985) no. 2, p. 269-281 | DOI | MR | Zbl
[Tsu86] - “Multitensors of differential forms on the Hilbert modular variety and on its subvarieties”, Math. Ann. 274 (1986) no. 4, p. 659-670 | DOI | MR | Zbl
Cité par Sources :