Eremenko, A. A Picard type theorem for holomorphic curves. (English) Zbl 0940.32010 Period. Math. Hung. 38, No. 1-2, 39-42 (1999). 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 V. A. Babets [Sib. Math. J. 25, 195-200 (1984; Zbl 0579.32038)]; when \(H_j\) are hyperplanes it follows by a result of M. G. Zajdenberg [Sib. Math. J. 24, 858-867 (1983; Zbl 0579.32039)]. Reviewer: Nicolae Mihalache (Bucureşti) Cited in 1 ReviewCited in 7 Documents MSC: 32H25 Picard-type theorems and generalizations for several complex variables 32C25 Analytic subsets and submanifolds Keywords:Picard type theorem; holomorphic curves Citations:Zbl 0579.32038; Zbl 0579.32039 PDFBibTeX XMLCite \textit{A. Eremenko}, Period. Math. Hung. 38, No. 1--2, 39--42 (1999; Zbl 0940.32010) Full Text: DOI