Picard-type theorems for holomorphic mappings
The author proves Picard-type theorems about the degeneracy of holomorphic mappings at a point. He essentially uses the relation with logarithmic differentials. Let W be a compact complex algebraic variety, D a divisor on W with normal intersection, and \(H^ 0(T^*(\log D))\) the space of D- logarithmically meromorphic 1-forms on W [see \textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 40, 5-57 (1971; Zbl 0219.14007)]. Theorem 1. Let W and D be as above and \(f: {\mathbb{C}}\to W\setminus D\) be a nonconstant holomorphic mapping. Then there exists a nonconstant holomorphic mapping \(\check f: {\mathbb{C}}\to W,\) on which all differentials \(\omega_ j\in H^ 0(T^*(\log D))\) take the constant values \((\omega_ j,\check f')=c_ j\), where \(c_ j\) are constants and \(\check f': {\mathbb{C}}\to T(W)\) is the derivative of the curve \v{f}. As an application of Theorem 1, the author proves the following theorem. Theorem 2. Let \(D_ j\) be hypersurfaces of certain degrees in \(CP_ n\) and \(D=\sum^{2n+1}_{j=1}D_ j\) be a divisor with normal intersection. If \(f: C\to CP_ n\setminus D\) is a holomorphic mapping, then f is equal to a constant.
degeneracy of holomorphic mappings at a point
scientific article; zbMATH DE number 3927367