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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0308433-6
MathSciNet review: 0308433
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0308433-6
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