Uniqueness problem of meromorphic mappings in several complex variables for moving targets
The author gives some uniqueness theorems for meromorphic mappings from \(\mathbb{C}^m\) into complex projective spaces and for moving hyperplanes (moving targets). For entire or meromorphic functions of several complex variables, \textit{B.-Q. Li} [Am. J. Math. 119, No. 4, 841-858 (1997; Zbl 0884.32002)] and \textit{M. Shirosaki} [Tohoku Math. J., II. Ser. 43, No. 3, 355-360 (1991; Zbl 0774.32005)] obtained some uniqueness theorems in the case of moving targets. On the other hand, in the case of fixed hyperplanes, some uniqueness theorems were obtained by \textit{H. Fujimoto} [Nagoya Math. J. 58, 1-23 (1975; Zbl 0313.32005)] and \textit{L. Smiley} [Contemp. Math. 25, 149-154 (1983; Zbl 0571.32002)]. Those results are extensions of the famous uniqueness theorem for meromorphic functions of one complex variable due to R. Nevanlinna. In this paper the author gives an extention of theorems due to H. Fujimoto and L. Smiley. The main result can be stated as follows: Let \(f,g:\mathbb{C}^m\to \mathbb{P}_N(\mathbb{C})\) be noncompact meromorphic mappings and \(a_j: \mathbb{C}^m\to \mathbb{P}_N(\mathbb{C})\) meromorphic mappings in general position that are small with respect to \(f\), where \(1\leq j\leq q\). Assume that \(f\) is linearly nondegenerate over the field \(K\) generated by components of all \(a_j\). Suppose that \((f,a_j)\) and \((g,a_j)\) have the same zero divisor (say \(Z_j\)) and \(\dim Z_i\cap Z_j\leq m-2\) for \(i\neq j\). We also assume that \(f= g\) on \(\bigcup^q_{j=1} Z_j\). Then \(f= Lg\) for some matrix \(L\) with elements in \(K\) if \(q= 3N+1\). If \(q= 3N+2\) and \(g\) is also linearly nondegenerate over the field \(K\), then \(f= g\). In the proofs, the author uses a second main theorem for meromorphic mappings and for moving targets due to \textit{M. Ru} and \textit{W. Stoll} [J. Geom. Anal. 1, No. 2, 99-138 (1991; Zbl 0732.30025)], and the idea due to H. Fujimoto and M. Shirosaki in the above cited papers.
uniqueness theorem for meromorphic functions
scientific article; zbMATH DE number 1911569
scientific article