Abstract
The classical little Picard theorem states that every entire function f missing two values must be constant. In (3.10.2) we stated E. Borel’s generalization to a system of entire functions. One of its geometric consequences is that given n + p hyperplanes H 1,..., H n +p in P n C in general position, every \(f \in Hol(C,{P_n}C - \cup _{i = 1}^{n + p}{H_i})\) has its image in a linear subspace of dimension ≤ [n/p], see (3.10.7). For p = n + 1, this means that \(f \in Hol(C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i})\) must be constant, (see (3.10.8)). This has been further strengthened to the statement that \({P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\) is complete hyperbolic and is hyperbolically imbedded in P n C, (see (3.10.9)), which is a statement on Hol(\(D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)) rather than on Hol(\(C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)) since the hyperbolicity and hyperbolic imbeddedness of \(X = {P_2}C - \cup _{i = 1}^{2n + 1}{H_i}\) is defined in terms of d X which, in turn, is constructed by Hol(\(D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)). Thus, (3.10.9) is a statement en termes finis in the sense of Bloch [1].
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© 1998 Springer-Verlag Berlin Heidelberg
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Kobayashi, S. (1998). Extension and Finiteness Theorems. In: Hyperbolic Complex Spaces. Grundlehren der mathematischen Wissenschaften, vol 318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03582-5_6
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DOI: https://doi.org/10.1007/978-3-662-03582-5_6
Publisher Name: Springer, Berlin, Heidelberg
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