×

zbMATH — the first resource for mathematics

Hyperbolicity of geometric orbifolds. (English) Zbl 1196.32018
The author studies the hyperbolic aspects of the geometric orbifolds, which were introduced by F. Campana [Ann. Inst. Fourier 54, No. 3, 499–630 (2004; Zbl 1062.14014)]. Especially, the author deals with the hyperbolicity of higher-dimensional orbifolds. In the higher-dimensional case, the author proves that the classical and the non-classical hyperbolicity are different unlike in the case of dimension one. Furthermore, the author defines algebraic hyperbolicity in the orbifold setting, which contains previous definitions of algebraic hyperbolicity in the compact and logarithmic settings, and by using it, the author generalizes the Kobayashi conjecture [see Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften. 318. Berlin: Springer (1998; Zbl 0917.32019)]. Even though the author does not prove this conjecture, he gets some results towards this conjecture and proves the following theorem (one of the main results of this paper) by using orbifold symmetric differentials.
Theorem. For a given smooth projective orbifold surface of general type with some hypotheses, there exists a proper subvariety \(Y\) of the smooth projective surface \(X\) such that every entire curve \(f:\mathbb C\to X\) satisfies \(f(\mathbb C) \subset Y\).
Finally, the author studies the orbifold measure hyperbolicity and generalizes some previous results.

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14D06 Fibrations, degenerations in algebraic geometry
32H30 Value distribution theory in higher dimensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. A. Bogomolov, Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041 – 1044 (Russian).
[2] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227 – 1287, 1439 (Russian).
[3] Fedor Bogomolov and Yuri Tschinkel, Special elliptic fibrations, The Fano Conference, Univ. Torino, Turin, 2004, pp. 223 – 234. · Zbl 1069.14009
[4] Marco Brunella, Courbes entières et feuilletages holomorphes, Enseign. Math. (2) 45 (1999), no. 1-2, 195 – 216 (French). · Zbl 1004.32011
[5] Frédéric Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499 – 630 (English, with English and French summaries). · Zbl 1062.14014
[6] F. Campana, Orbifoldes spéciales et classification biméromorphe des variétés kählériennes compactes, arXiv:0705.0737. · Zbl 1236.14039
[7] Frédéric Campana and Mihai Păun, Variétés faiblement spéciales à courbes entières dégénérées, Compos. Math. 143 (2007), no. 1, 95 – 111 (French, with English summary). · Zbl 1120.32013
[8] F. Campana, J. Winkelmann, A Brody theorem for orbifolds, preprint 2006, arXiv: math/0604571. · Zbl 1162.14012
[9] James Carlson and Phillip Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557 – 584. · Zbl 0248.32018
[10] Xi Chen, On algebraic hyperbolicity of log varieties, Commun. Contemp. Math. 6 (2004), no. 4, 513 – 559. · Zbl 1083.14052
[11] Jean-Pierre Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285 – 360. · Zbl 0919.32014
[12] Gerd-Eberhard Dethloff and Steven Shin-Yi Lu, Logarithmic jet bundles and applications, Osaka J. Math. 38 (2001), no. 1, 185 – 237. · Zbl 0982.32022
[13] Jawher El Goul, Logarithmic jets and hyperbolicity, Osaka J. Math. 40 (2003), no. 2, 469 – 491. · Zbl 1048.32016
[14] Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. · Zbl 0764.30001
[15] Alessandro Ghigi and János Kollár, Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007), no. 4, 877 – 902. · Zbl 1135.53031
[16] Mark Green and Phillip Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 41 – 74. · Zbl 0508.32010
[17] J. P. Jouanolou, Hypersurfaces solutions d’une équation de Pfaff analytique, Math. Ann. 232 (1978), no. 3, 239 – 245 (French). · Zbl 0354.34007
[18] Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. · Zbl 0917.32019
[19] Shoshichi Kobayashi and Takushiro Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7 – 16. · Zbl 0331.32020
[20] Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. · Zbl 0628.32001
[21] Michael McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121 – 174. · Zbl 1006.32020
[22] M. McQuillan, Noncommutative Mori theory, preprint IHES (2000).
[23] M. McQuillan, Bloch hyperbolicity, preprint IHES (2001).
[24] M. McQuillan, Rational criteria for hyperbolicity, Book preprint.
[25] Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.12501
[26] E. I. Nochka, On the theory of meromorphic curves, Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 547 – 552 (Russian). · Zbl 0552.32024
[27] Gianluca Pacienza and Erwan Rousseau, On the logarithmic Kobayashi conjecture, J. Reine Angew. Math. 611 (2007), 221 – 235. · Zbl 1133.32015
[28] Erwan Rousseau, Hyperbolicité du complémentaire d’une courbe dans \Bbb P²: le cas de deux composantes, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 635 – 640 (French, with English and French summaries). · Zbl 1034.32017
[29] Fumio Sakai, Degeneracy of holomorphic maps with ramification, Invent. Math. 26 (1974), 213 – 229. · Zbl 0276.32012
[30] A. Seidenberg, Reduction of singularities of the differential equation \?\?\?=\?\?\?, Amer. J. Math. 90 (1968), 248 – 269. · Zbl 0159.33303
[31] B. V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs, vol. 61, American Mathematical Society, Providence, RI, 1985. Translated from the Russian by J. R. King; Translation edited by Lev J. Leifman. · Zbl 0564.32016
[32] Y.T. Siu, A proof of the generalized Schwarz lemma using the logarithmic derivative lemma. Private communication to J.-P. Demailly, Journal de la SMF (1997).
[33] Paul Vojta, On the \?\?\? conjecture and Diophantine approximation by rational points, Amer. J. Math. 122 (2000), no. 4, 843 – 872. · Zbl 1037.11052
[34] P.-M. Wong, Nevanlinna theory for holomorphic curves in projective varieties, preprint (1999).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.