A note on the unicity of holomorphic curves in \(\mathbb{P}^2 (\mathbb{C})\). (Chinese. English summary) Zbl 07802329

Han, Jing; Shen, Jiaji

Chin. Ann. Math., Ser. A 44, No. 3, 315-322 (2023).

Summary: In this paper, the unicity of holomorphic curves in \(\mathbb{P}^2 (\mathbb{C})\) intersecting hyperplanes in subgeneral position is discussed. Let \(f_1\), \(f_2, \cdots, f_\lambda\) be linearly non-degenerate holomorphic curves in \(\mathbb{P}^2 (\mathbb{C})\), and \(H_1\), \(H_1, \cdots, H_q\) be hyperplanes in \(\mathbb{P}^2 (\mathbb{C})\) located in \(m\)-subgeneral position such that \(A_j :=f_1^{-1} (H_j) = \cdots =f_\lambda^{-1} (H_j)\) for \(j=1,\cdots,q\), and \(A_i \cap A_j = \emptyset\) for \((i \neq j)\). Assume that there exists an integer \(l\) with \(2 \leqslant l \leqslant \lambda\) such that \(f_{j_1} (z) \wedge f_{j_2} (z) \wedge \cdots \wedge f_{j_l} (z) = 0 (z\in \bigcup\limits^q_{j=1} A_j)\) for any \(l\) indices \(1 \leqslant j_1 < j_2 < \cdots < j_l \leqslant \lambda\). Then, when \(q > \frac{2\lambda}{\lambda -l+1} + \frac{3}{2}m\), \(f_1 \wedge \cdots \wedge f_\lambda \equiv 0\). The key technique is an improved second main theorem with inequality \(\| (q -\frac{3m}{2}) T_{f_t} (r) \leqslant \sum\limits^q_{j=1} N^2_{(f_t, H_j)} (r,0) + o(T_{f_t} (r))\) for \(t=1,\cdots, \lambda\).

MSC:

32H30	Value distribution theory in higher dimensions
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

holomorphic curve; hyperplane; second main theorem; uniqueness problem; subgeneral position

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