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Meromorphic mappings and deficiencies. (English) Zbl 1066.32023
Miyajima, Kimio (ed.) et al., Complex analysis in several variables—Memorial conference of Kiyoshi Oka’s centennial birthday. Papers from the conference, Kyoto, Japan, October 30–November 5, 2001 and Nara, Japan, November 6–8, 2001. Tokyo: Mathematical Society of Japan (ISBN 4-931469-27-2/hbk). Advanced Studies in Pure Mathematics 42, 237-242 (2004).
The author exhibited two elimination theorems of defects of hypersurfaces or rational moving targets for a meromorphic mapping or a holomorphic curve into \(\mathbb{P}^n(\mathbb{C})\) by its small deformation. One of them is the following: Let \(f:\mathbb{C}^m\rightarrow\mathbb{P}^n(\mathbb{C})\) be a transcendental meromorphic mapping, and \(d\) is a positive integer. Then there exists a regular matrix \(L=(l_{ij})_{0\leq i,j\leq n}\) of the form \(l_{ij}=c_{ij}g_j+d_{ij}\;(c_{ij},d_{ij}\in \mathbb{C}: 0\leq i,j\leq n)\) such that \(\det L\not=0\) and \(\widetilde{f}=L\cdot f:\mathbb{C}^m\rightarrow\mathbb{P}^n(\mathbb{C})\) is a meromorphic mapping without Nevanlinna defects of hypersurfaces of degree at most \(d\), and satisfies \[ | T_f(r)-T_{\widetilde{f}}(r)| =O(\log r),\;r\to\infty, \] where \(g_j\;(j=1,\dots,n)\) are some monomials on \(\mathbb{C}^m\).
The author has proved the case of hyperplanes before [S. Mori, Ann. Acad. Sci. Fenn. 24, 89-104 (1999; Zbl 0930.32011)].
For the entire collection see [Zbl 1050.32002].
Reviewer: Pei-Chu Hu (Jinan)
32H30 Value distribution theory in higher dimensions
32H04 Meromorphic mappings in several complex variables