On holomorphic curves extremal for the truncated defect relation and some applications
type: Item
instance of: scholarly article
title: On holomorphic curves extremal for the truncated defect relation and some applications
zbMATH Open document ID: 1088.32006
author: Nobushige Toda
published in: Proceedings of the Japan Academy. Series A
publication date: 2006-02-21T00:00:00Z
review text: The paper under review deals with extremal holomorphic curves for the truncated defect relation. Let \(f:\mathbb C\to \mathbb P_n(\mathbb C)\) be a transcendental linearly nondegenerate holomorphic curve and \(X\) a subset of \(\mathbb C^{n+1}\setminus \{0\}\) in \(N\)-subgeneral position. Denote by \(\delta_n({\mathbf a},f)\) the \(n\)-truncated defect of \(f\) at \(\mathbf a\). We define \(D^+_n(X,f) = \{\mathbf a\in X|\delta_n(\mathbf a,f) > 0\}\) and \(D^1_n(X, f) = \{\mathbf a\in X|\delta_n(\mathbf a,f) =1\}\). If there are \(n+1\) linearly independent vectors in \(D^1_n(X,f)\), then \(\#D^+_n(X,f)\leq (n + 1)(N + 1-n)\) (Theorem 3.1). The main purpose in this paper is to give estimate for the cardinality of \(D^1_n(X,f)\) when \(f\) is extremal for the truncated defect relation. Suppose that \(\sum_{\mathbf a_j\in D^1_n(X,f)}\delta(\mathbf a_j,f)= 2N-n+1\). If there are \(n + 1\) linearly independent vectors in \(D^1_n(X,f)\), then \(D^+_n(X,f)=D^1_n(X,f)\) and \(\#D^1_n(X,f)=2N-n + 1\) (Theorem 3.2). When there are \(n\) linearly independent vectors in \(D^1_n(X,f)\), if \(\# D^1_n(X,f) < 2N-n+1\), then \(\#D^1_n(X,f) = N\) (Theorem 3.3). The author also considers another defect and obtains the results similar to the above.
reviewed by: Yoshihiro Aihara
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5008560
zbMATH Keywords: holomorphic curves
zbMATH Keywords: truncated defect
MaRDI profile type: Publication
full work available at URL: pjaa.81.99
OpenAlex ID: W1998841299
cites work: Q4753437
cites work: Théorie nouvelle des familles complexes normales. Applications à l'étude des fonctions algébroïdes
cites work: Q4834182
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cites work: On a holomorphic curve extremal for the defect relation
DOI: 10.3792/PJAA.81.99
description: scientific article; zbMATH DE number 5008560