Degeneracy of holomorphic maps via orbifolds
Bulletin de la Société Mathématique de France, Volume 140 (2012) no. 4, p. 459-484

We use orbifold structures to deduce degeneracy statements for holomorphic maps into logarithmic surfaces. We improve former results in the smooth case and generalize them to singular pairs. In particular, we give applications on nodal surfaces and complements of singular plane curves.

Nous utilisons les structures orbifoldes pour obtenir des résultats de dégénérescence des applications holomorphes dans les surfaces logarithmiques. Nous améliorons certains résultats déjà obtenus dans le cas lisse et les généralisons aux paires singulières. En particulier, nous illustrons nos résultats sur les surfaces nodales et les complémentaires de courbes planes singulières.

DOI : https://doi.org/10.24033/bsmf.2633
Classification:  32H30,  32Q45,  14J29
Keywords: entire curves, Kobayashi hyperbolicity, orbifolds
@article{BSMF_2012__140_4_459_0,
     author = {Rousseau, Erwan},
     title = {Degeneracy of holomorphic maps via orbifolds},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {140},
     number = {4},
     year = {2012},
     pages = {459-484},
     doi = {10.24033/bsmf.2633},
     zbl = {1271.32019},
     mrnumber = {3059847},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2012__140_4_459_0}
}
Degeneracy of holomorphic maps via orbifolds. Bulletin de la Société Mathématique de France, Volume 140 (2012) no. 4, pp. 459-484. doi : 10.24033/bsmf.2633. http://www.numdam.org/item/BSMF_2012__140_4_459_0/

[1] D. Abramovich & A. Vistoli - « Compactifying the space of stable maps », J. Amer. Math. Soc. 15 (2002), p. 27-75. | MR 1862797 | Zbl 0991.14007

[2] A. Adem, J. Leida & Y. Ruan - Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge Univ. Press, 2007. | MR 2359514 | Zbl 1157.57001

[3] F. Bogomolov - « Families of curves on a surface of general type », Sov. Math. Dokl. 18 (1977), p. 1294-1927. | Zbl 0415.14013

[4] -, « Holomorphic tensors and vector bundles on projective varieties », Math. USSR Izvestija 13 (1979), p. 499-555. | Zbl 0439.14002

[5] F. Bogomolov & B. De Oliveira - « Hyperbolicity of nodal hypersurfaces », J. reine angew. Math. 596 (2006), p. 89-101. | MR 2254806 | Zbl 1108.14013

[6] F. Campana - « Orbifolds, special varieties and classification theory », Ann. Inst. Fourier (Grenoble) 54 (2004), p. 499-630. | Numdam | MR 2097416 | Zbl 1062.14014

[7] -, « Orbifoldes spéciales et classification biméromorphe des variétés kählériennes compactes », preprint arXiv:0705.0737. | Zbl 1236.14039

[8] F. Campana & M. Păun - « Variétés faiblement spéciales à courbes entières dégénérées », Compos. Math. 143 (2007), p. 95-111. | MR 2295198 | Zbl 1120.32013

[9] F. Campana & J. Winkelmann - « A Brody theorem for orbifolds », Manuscripta Math. 128 (2009), p. 195-212. | MR 2471315 | Zbl 1162.14012

[10] L. A. Campbell, A. Howard & T. Ochiai - « Moving holomorphic disks off analytic subsets », Proc. Amer. Soc. 60 (1976), p. 106-108. | MR 425186 | Zbl 0314.32014

[11] L. A. Campbell & R. H. Ogawa - « On preserving the Kobayashi pseudodistance », Nagoya Math. J. 57 (1975), p. 37-47. | MR 372258 | Zbl 0312.32014

[12] J. A. Carlson & M. Green - « A Picard theorem for holomorphic curves in the plane », Duke Math. J. 43 (1976), p. 1-9. | MR 397026 | Zbl 0333.32022

[13] P. Deligne & G. D. Mostow - Commensurabilities among lattices in PU (1,n), Annals of Math. Studies, vol. 132, Princeton Univ. Press, 1993. | MR 1241644 | Zbl 0826.22011

[14] G.-E. Dethloff & S. S.-Y. Lu - « Logarithmic jet bundles and applications », Osaka J. Math. 38 (2001), p. 185-237. | MR 1824906 | Zbl 0982.32022

[15] H. M. Farkas & I. Kra - Riemann surfaces, Graduate Texts in Math., vol. 71, Springer, 1980. | MR 583745 | Zbl 0475.30001

[16] A. Ghigi & J. Kollár - « Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres », Comment. Math. Helv. 82 (2007), p. 877-902. | MR 2341843 | Zbl 1135.53031

[17] H. Grauert & U. Peternell - « Hyperbolicity of the complement of plane curves », Manuscripta Math. 50 (1985), p. 429-441. | MR 784151 | Zbl 0581.32031

[18] M. Green & P. Griffiths - « Two applications of algebraic geometry to entire holomorphic mappings », in The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, 1980, p. 41-74. | MR 609557 | Zbl 0508.32010

[19] T. Kawasaki - « The Riemann-Roch theorem for complex V-manifolds », Osaka J. Math. 16 (1979), p. 151-159. | MR 527023 | Zbl 0405.32010

[20] R. Kobayashi - « Uniformization of complex surfaces », in Kähler metric and moduli spaces, Adv. Stud. Pure Math., vol. 18, Academic Press, 1990, p. 313-394. | MR 1145252 | Zbl 0755.32024

[21] S. Kobayashi - Hyperbolic complex spaces, Grundl. Math. Wiss., vol. 318, Springer, 1998. | MR 1635983 | Zbl 0917.32019

[22] R. Lazarsfeld - Positivity in algebraic geometry. II, Ergebn. Math. Grenzg., vol. 49, Springer, 2004. | MR 2095472 | Zbl 1093.14500

[23] K. Matsuki - Introduction to the Mori program, Universitext, Springer, 2002. | MR 1875410 | Zbl 0988.14007

[24] K. Matsuki & M. Olsson - « Kawamata-Viehweg vanishing as Kodaira vanishing for stacks », Math. Res. Lett. 12 (2005), p. 207-217. | MR 2150877 | Zbl 1080.14023

[25] M. Mcquillan - « Diophantine approximations and foliations », Publ. Math. I.H.É.S. 87 (1998), p. 121-174. | MR 1659270 | Zbl 1006.32020

[26] -, « A toric extension of Faltings' ‘Diophantine approximation on Abelian varieties' », J. Differential Geom. 57 (2001), p. 195-231. | MR 1879225 | Zbl 1070.11028

[27] -, « Rational criteria for hyperbolicity », preprint.

[28] G. Megyesi - « Generalisation of the Bogomolov-Miyaoka-Yau inequality to singular surfaces », Proc. London Math. Soc. 78 (1999), p. 241-282. | MR 1665244 | Zbl 1006.14012

[29] I. Moerdijk & J. Mrčun - Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Math., vol. 91, Cambridge Univ. Press, 2003. | MR 2012261 | Zbl 1029.58012

[30] R. Nevanlinna - Analytic functions, Grundl. Math. Wiss., vol. 162, Springer, 1970. | MR 279280 | Zbl 0199.12501

[31] E. Rousseau - « Hyperbolicity of geometric orbifolds », Trans. Amer. Math. Soc. 362 (2010), p. 3799-3826. | MR 2601610 | Zbl 1196.32018

[32] I. Satake - « The Gauss-Bonnet theorem for V-manifolds », J. Math. Soc. Japan 9 (1957), p. 464-492. | MR 95520 | Zbl 0080.37403

[33] Y. T. Siu - « A proof of the generalized Schwarz lemma using the logarithmic derivative lemma », private communication to J.-P. Demailly, 1997.

[34] B. Toen - « Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford », K-Theory 18 (1999), p. 33-76. | MR 1710187 | Zbl 0946.14004

[35] H.-H. Tseng - « Orbifold quantum Riemann-Roch, Lefschetz and Serre », Geom. Topol. 14 (2010), p. 1-81. | MR 2578300 | Zbl 1178.14058

[36] A. M. Uludağ - « Orbifolds and their uniformization », in Arithmetic and geometry around hypergeometric functions, Progr. Math., vol. 260, Birkhäuser, 2007, p. 373-406. | MR 2306159 | Zbl 1126.32020

[37] P. M. Wong - « Nevanlinna theory for holomorphic curves in projective varieties », preprint, 1999.