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The hyperbolicity of the complement of $ 2n+1$ hyperplanes in general position in $ P\sb{n}$ and related results


Author: Mark L. Green
Journal: Proc. Amer. Math. Soc. 66 (1977), 109-113
MSC: Primary 32H20
DOI: https://doi.org/10.1090/S0002-9939-1977-0457790-9
MathSciNet review: 0457790
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Abstract | References | Similar Articles | Additional Information

Abstract: Using a modified version of a technique of R. Brody, a simple proof is found that the complement of 2n + 1 hyperplanes in general position in $ {{\mathbf{P}}_n}$ is complete hyperbolic and hyperbolically embedded in $ {{\mathbf{P}}_n}$. In fact, a more general result is obtained showing that a suitable Picard theorem is sufficient to imply hyperbolicity in a large class of algebro-geometric situations.


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

  • [1] A. Bloch, Sur les systèmes de fonctions holormorphes à variétés linéaires lacunaires, Ann. Sci. École Norm Sup 43 (1926), 309-362. MR 1509274
  • [2] R. Brody, Intrinsic metrics and measures on compact complex manifolds, Thesis, Harvard Univ., Cambridge, Mass.
  • [3] H. Cartan, Sur les systèmes de fonctions holomorphes à variétés lacunaires et leurs applications, Ann. Sci. École Norm. Sup. 45 (1928), 255-346. MR 1509288
  • [4] M. Cowen, The Kobayashi metric on $ {{\mathbf{P}}_n} - ({2^n} + 1)$ hyperplanes, Value-Distribution Theory, Part A, Dekker, New York, 1974, pp. 205-224. MR 0352543 (50:5030)
  • [5] M. Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43-75. MR 0367302 (51:3544)
  • [6] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Dekker, New York, 1970. MR 0277770 (43:3503)
  • [7] R. Nevanlinna, Analytic functions, Springer-Verlag, Berlin, 1970. MR 0279280 (43:5003)
  • [8] H. L. Royden, Remarks on the Kobayashi metric, Lecture Notes in Math., no. 185, Springer-Verlag, Berlin and New York, 1971, pp. 125-137. MR 0304694 (46:3826)
  • [9] P. Kiernan and S. Kobayashi, Holomorphic mappings into protective space with lacunary hyperplanes, Nagoya Math. J. 50 (1973), 199-216. MR 0326007 (48:4353)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0457790-9
Keywords: Hyperbolic manifold, holomorphic curve, value distribution theory, Picard theorem, hyperplanes in general position, complex projective spaces
Article copyright: © Copyright 1977 American Mathematical Society

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