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**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.

Reviewer: Yoshihiro Aihara (Shizuoka)

### MSC:

32H30 | Value distribution theory in higher dimensions |

32H25 | Picard-type theorems and generalizations for several complex variables |

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\textit{L. Jin} and \textit{M. Ru}, Tohoku Math. J. (2) 57, No. 4, 589--595 (2005; Zbl 1106.32017)

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### References:

[1] | H. Fujimoto, The uniqueness problem of meromorphic maps into the complex projective space, Nogoya Math. J. 58 (1975), 1–23. · Zbl 0313.32005 |

[2] | H. Fujimoto, Value distribution theory of the Gauss map of minimal surfaces in \(\mathbb R^m\), Aspects Math. E21, Vieweg, Braunschweing, 1993. · Zbl 1107.32004 |

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[8] | Z.-H. Tu, Uniquenness problem of meromorphic mappings in several complex variables for moving targets, Tohoku Math. J. (2) 54 (2002), 567–579. · Zbl 1027.32017 |

[9] | J. T.-Y. Wang, A generalization of Picard’s theorem with moving targets, Complex Var. Theory Appl. 44 (2001), 39–45. · Zbl 1026.32037 |

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