×

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.

MSC:

32H30 Value distribution theory in higher dimensions
32H25 Picard-type theorems and generalizations for several complex variables
PDF BibTeX XML Cite
Full Text: DOI

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
[3] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. · Zbl 0628.32001
[4] M. Ru, Nevanlinna theory and its relation to Diophantine approximation, World Scientific Publishing Co., Singapore, 2001. · Zbl 0998.30030
[5] M. Ru and W. Stoll, The second main theorem for moving targets, J. Geom. Anal. 1 (1991), 99–138. · Zbl 0732.30025
[6] M. Ru and W. Stoll, The Cartan conjecture for moving targets, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 477–508, Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, R.I., 1991. · Zbl 0742.32019
[7] M. Shirosaki, An extension of unicity theorem for meromorphic functions, Tohoku Math. J. (2) 45 (1993), 491–497. · Zbl 0802.30026
[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
[10] Z. Ye, A unicity theorem for meromorphic mappings, Houston J. Math. 24 (1998), 519–531. · Zbl 0971.32008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.