On holomorphic curves extremal for the truncated defect relation.
Let \(f\) be a transcendental linearly non-degenerate holomorphic curve from \(\mathbb C\) to the \(n\)-dimen\-sional complex projective space \(P^n\mathbb C\). Denote by \(\delta_n(\mathbf{a}, f)\) the truncated deficiency of \(\mathbf{a}\in \mathbb C^{n+1}\setminus \{\mathbf{0}\}\) and by \(\mathbf{X}\) a subset of \(\mathbb C^{n+1}\setminus \{\mathbf{0}\}\) in \(N\)-subgeneral position with \(N\geq n\). The author in this paper proves that if \(N>n=2m-1\) and there exists an infinite number of vectors \(a_j\in X\) satisfying \(\delta_n(a_j, f)>0\) and \(\sum_{j=1}^{+\infty}\delta_n(a_j, f)=2N-n+1\), then, either \(\#\{j:\delta_n(a_j,f)=1\}>(2N-n+1)/(n+1)\); or, there exist mutually disjoint subsets \(M_k\) of \(\mathbb{N}\) such that \(\bigcup_{k=1}^{+\infty} M_k = \mathbb{N}\), \(\#M_k= N-m+1, d(M_k)=m\) and any \(m\) elements of \(\{a_j:j\in\mathbb{N}\}\) are linearly independent. Note that the author proved a similar theorem for the case of \(n=2m\) in [Bull. Nagoya Inst. Technol. 55, 1--18 (2003; Zbl 1076.32502)].
scientific article; zbMATH DE number 5123004