Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hyperbolicity of geometric orbifolds


Author: Erwan Rousseau
Journal: Trans. Amer. Math. Soc. 362 (2010), 3799-3826
MSC (2000): Primary 14D06, 32H30, 32Q45
Published electronically: February 24, 2010
MathSciNet review: 2601610
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study complex hyperbolicity in the setting of geometric orbifolds introduced by F. Campana. Generalizing classical methods to this context, we obtain degeneracy statements for entire curves with ramification in situations where no Second Main Theorem is known from value distribution theory.


References [Enhancements On Off] (What's this?)

  • 1. F. A. Bogomolov, Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041–1044 (Russian). MR 0457450
  • 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). MR 522939
  • 3. Fedor Bogomolov and Yuri Tschinkel, Special elliptic fibrations, The Fano Conference, Univ. Torino, Turin, 2004, pp. 223–234. MR 2112577
  • 4. Marco Brunella, Courbes entières et feuilletages holomorphes, Enseign. Math. (2) 45 (1999), no. 1-2, 195–216 (French). MR 1703368
  • 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). MR 2097416
  • 6. F. Campana, Orbifoldes spéciales et classification biméromorphe des variétés kählériennes compactes, arXiv:0705.0737.
  • 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). MR 2295198, 10.1112/S0010437X06002685
  • 8. F. Campana, J. Winkelmann, A Brody theorem for orbifolds, preprint 2006, arXiv: math/0604571.
  • 9. James Carlson and Phillip Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557–584. MR 0311935
  • 10. Xi Chen, On algebraic hyperbolicity of log varieties, Commun. Contemp. Math. 6 (2004), no. 4, 513–559. MR 2078413, 10.1142/S0219199704001422
  • 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. MR 1492539, 10.1090/pspum/062.2/1492539
  • 12. Gerd-Eberhard Dethloff and Steven Shin-Yi Lu, Logarithmic jet bundles and applications, Osaka J. Math. 38 (2001), no. 1, 185–237. MR 1824906
  • 13. Jawher El Goul, Logarithmic jets and hyperbolicity, Osaka J. Math. 40 (2003), no. 2, 469–491. MR 1988702
  • 14. Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
  • 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. MR 2341843, 10.4171/CMH/113
  • 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. MR 609557
  • 17. J. P. Jouanolou, Hypersurfaces solutions d’une équation de Pfaff analytique, Math. Ann. 232 (1978), no. 3, 239–245 (French). MR 0481129
  • 18. Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983
  • 19. Shoshichi Kobayashi and Takushiro Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7–16. MR 0402127
  • 20. Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677
  • 21. Michael McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121–174. MR 1659270
  • 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. MR 0279280
  • 26. E. I. Nochka, On the theory of meromorphic curves, Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 547–552 (Russian). MR 701289
  • 27. Gianluca Pacienza and Erwan Rousseau, On the logarithmic Kobayashi conjecture, J. Reine Angew. Math. 611 (2007), 221–235. MR 2361090, 10.1515/CRELLE.2007.080
  • 28. Erwan Rousseau, Hyperbolicité du complémentaire d’une courbe dans ℙ²: le cas de deux composantes, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 635–640 (French, with English and French summaries). MR 1988123, 10.1016/S1631-073X(03)00136-5
  • 29. Fumio Sakai, Degeneracy of holomorphic maps with ramification, Invent. Math. 26 (1974), 213–229. MR 0355118
  • 30. A. Seidenberg, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Amer. J. Math. 90 (1968), 248–269. MR 0220710
  • 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. MR 807367
  • 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 𝐴𝐵𝐶 conjecture and Diophantine approximation by rational points, Amer. J. Math. 122 (2000), no. 4, 843–872. MR 1771576
  • 34. P.-M. Wong, Nevanlinna theory for holomorphic curves in projective varieties, preprint (1999).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14D06, 32H30, 32Q45

Retrieve articles in all journals with MSC (2000): 14D06, 32H30, 32Q45


Additional Information

Erwan Rousseau
Affiliation: Département de Mathématiques, IRMA, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg Cedex, France
Email: rousseau@math.u-strasbg.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05019-1
Keywords: Orbifolds, Kobayashi hyperbolicity, entire curves
Received by editor(s): September 15, 2008
Published electronically: February 24, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.