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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

General defect relations of holomorphic curves


Author: Kiyoshi Niino
Journal: Trans. Amer. Math. Soc. 289 (1985), 99-113
MSC: Primary 30D35; Secondary 32H30
MathSciNet review: 779054
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Abstract: Let $ x:{\mathbf{C}} \to {P_n}{\mathbf{C}}$ be a holomorphic curve of finite lower order $ \mu $, and let $ A = \{ \alpha \} $ be an arbitrary finite family of holomorphic curves $ \alpha :{\mathbf{C}} \to {({P_n}{\mathbf{C}})^\ast}$ satisfying $ T(r,\alpha ) = o(T(r,x))\;(r \to \infty )$. Suppose $ x$ is nondegenerate with respect to $ A$, and $ A$ is in general position. We show the following general defect relations: (1) $ x$ has at most $ n$ deficient curves in $ A$ if $ \mu = 0$. (2) $ \sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq n\;{\text{if}}\;0 < \mu \leq 1/2} $. (3) $ \sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text{if}}\;1/2 < \mu < + \infty } $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0779054-5
PII: S 0002-9947(1985)0779054-5
Keywords: Holomorphic curve, Nevanlinna theory, defect relation, spread relation, deficient curve
Article copyright: © Copyright 1985 American Mathematical Society