A unicity theorem for moving targets counting multiplicities. (English) Zbl 1106.32017

The authors give uniqueness theorems for holomorphic curves \(f: \mathbb{C}\to\mathbb{P}_N(\mathbb{C})\) and for moving hyperplanes, which are an extension of the uniqueness theorems due to H. Fujimoto [Nagoya Math. J. 58, 1–23 (1975; Zbl 0313.32005)] and Z. H. Tu [Tohoku Math. J., II. Ser. 54, No. 4, 567–579 (2002; Zbl 1027.32017)]. The main result can be stated as follows: Let \(f,g: \mathbb{C}\to\mathbb{P}_N(\mathbb{C})\) be nonconstant holomorphic curves and \(H_j\) \((j= 1,\dots, q)\) moving hyperplanes in general position that are small with respect to \(f\) generated by coefficients of all \(H_j\). Suppose that \(f^*H_j\) and \(g^* H_j\) are identical as divisors. Then \(f= Lg\) for some matrix \(L\) with elements in \({\mathcal R}\) if \(q =3N+ 1\). If \(q= 3N+ 2\) and \(f\) is also linearly nondegenerate over the field \({\mathcal R}\), then \(f= g\). In the proofs, they use a Borel’s identity and the idea due to H. Fujimoto in the above cited paper.


32H30 Value distribution theory in higher dimensions
32H25 Picard-type theorems and generalizations for several complex variables
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