Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Holomorphic maps into complex projective space omitting hyperplanes

Author: Mark L. Green
Journal: Trans. Amer. Math. Soc. 169 (1972), 89-103
MSC: Primary 32A30; Secondary 32H25
MathSciNet review: 0308433
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using methods akin to those of E. Borel and R. Nevanlinna, a generalization of Picard's theorem to several variables is proved by reduction to a lemma on linear relations among exponentials of entire functions. More specifically, it is shown that a holomorphic map from $ {{\mathbf{C}}^m}$ to $ {{\mathbf{P}}_n}$ omitting $ n + 2$ distinct hyperplanes has image lying in a hyperplane. If the map omits $ n + 2$ or more hyperplanes in general position, the image will lie in a linear subspace of low dimension, being forced to be constant if the map omits $ 2n + 1$ hyperplanes in general position. The limits found for the dimension of the image are shown to be sharp.

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

  • [1] E. Borel, Sur les zéros des fonctions entières, Acta Math. 20 (1887), 357-396. MR 1554885
  • [2] James Carlson, Some degeneracy theorems for entire functions with values in an algebraic variety, Thesis, Princeton University, Princeton, N. J., 1971. MR 0296356 (45:5417)
  • [3] S. S. Chern, Proceedings International Congress of Mathematicians (Nice, 1970).
  • [4] J. Dufresnoy, Théorie nouvelle des famillies complexes normales. Applications à l'étude des fonctions algébroides, Ann. Sci. École Norm. Sup. (3) 61 (1944), 1-44. MR 7,289. MR 0014469 (7:289f)
  • [5] Peter Kiernan, Hyperbolic submanifolds of complex projective space, Proc. Amer. Math. Soc. 22 (1969), 603-606. MR 39 #7134. MR 0245828 (39:7134)
  • [6] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthier-Villars, Paris, 1929.
  • [7] H. Wu, The equidistribution theory of holomorphic curves, Princeton Univ. Press, Princeton, N. J., 1970. MR 0273070 (42:7951)
  • [8] -, An $ n$-dimensional extension of Picard's theorem, Bull. Amer. Math. Soc. 75 (1969), 1357-1361. MR 40 #7482. MR 0254273 (40:7482)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32A30, 32H25

Retrieve articles in all journals with MSC: 32A30, 32H25

Additional Information

Keywords: Picard theorem, value distribution theory, exponential function, Nevanlinna theory, hyperplanes in general position, holomorphic curve, Nevanlinna characteristic function
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society