A Picard type theorem for holomorphic curves
The following theorem is proved: Let \(M\) be a closed subset of \({\mathbb CP}^m\) and let \(\{H_j\}_{j=1}^{2n+1}\) be hypersurfaces such that \(M\cap(\bigcap_{j\in I}H_j)=\emptyset\) for every \(I\subset \{1,\cdots,2n+1\}\), \(|I|=n+1\). Then every holomorphic map \(f:{\mathbb C}\mapsto M\setminus(\bigcup_{j=1}^{2n+1}H_j)\) is constant. As a corollary one proves that, under the same hypothesis, if \(M\) is a projective subvariety of \({\mathbb CP}^m\) then \(M\setminus(\bigcup_{j=1}^{2n+1}H_j)\) is complete hyperbolic and hyperbolically embedded into \(M\). This last result was known for \(M={\mathbb CP}^m\) and \(m=n\) by a previous work of \textit{V. A. Babets} [Sib. Math. J. 25, 195-200 (1984; Zbl 0579.32038)]; when \(H_j\) are hyperplanes it follows by a result of \textit{M. G. Zajdenberg} [Sib. Math. J. 24, 858-867 (1983; Zbl 0579.32039)].
scientific article; zbMATH DE number 1404407