On holomorphic curves extremal for the truncated defect relation.
type: Item
instance of: scholarly article
title: On holomorphic curves extremal for the truncated defect relation.
zbMATH Open document ID: 1116.32010
author: Nobushige Toda
published in: Proceedings of the Japan Academy. Series A
publication date: 2007-02-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1141279059
review text: 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)].
reviewed by: Zhuan Ye
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5123004
zbMATH Keywords: holomorphic curve
zbMATH Keywords: truncated defect relation
zbMATH Keywords: extremal
MaRDI profile type: Publication
OpenAlex ID: W2023014203
cites work: Q4753437
cites work: Q4834182
cites work: Q3345078
cites work: On the deficiency of holomorphic curves with maximal deficiency sum
cites work: Q4810624
cites work: On holomorphic curves extremal for the truncated defect relation and some applications
cites work: Q5426456
DOI: 10.3792/PJAA.82.18
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description: scientific article; zbMATH DE number 5123004