Finiteness theorems for meromorphic mappings
type: Item
instance of: scholarly article
title: Finiteness theorems for meromorphic mappings
zbMATH Open document ID: 0926.32022
author: Yoshihiro Aihara
published in: Osaka Journal of Mathematics
publication date: 1999-11-18T00:00:00Z
review text: The main purpose of this paper is to prove finiteness theorems for some families of meromorphic mappings that are transcendental in general. Let \(L\to M\) be a fixed line bundle over \(M,\) and let \(\sigma_1,\dots,\sigma_s\) be linearly independent holomorphic sections of \(L \to M\) with \(s\geq 2.\) It is assumed that \((\sigma_j) = dD_j\) \((1 \leq j \leq s)\) for some positive integer \(d,\) where \(D_j\) are effective divisors on \(M.\) Set \(\varpi =c_1\sigma_1 +\dots+c_s\sigma_s,\) where \(c_j\in{\mathbb C}^\ast.\) Let \(D\) be a divisor defined by \(\varpi = 0.\) A meromorphic mapping \(\varPsi: M \to{\mathbb P}_{s-1}({\mathbb C})\) is defined by \(\varPsi=(\sigma_1,\dots,\sigma_s).\) Let \(p\) be a nonnegative integer. For a nonzero effective divisor \(E\) on \(\mathbb C^m,\) \(\mathcal F(p;(\mathbb C^m,E),(M,D))\) denotes the set of all meromorphic mappings \(f:\mathbb C^m\to M\) such that \(f^\ast D\equiv E\pmod p\). A meromorphic mapping \(f:\mathbb C^m\to M\) is said to be analytically nondegenerate if \(f(\mathbb C^m)\) is not included in any proper analytic subset of \(M.\) Let \(\mathcal F^\ast(p;(\mathbb C^m,E),(M,D))\) denote the subset of all \(f\in\mathcal F(p;(\mathbb C^m,E),(M,D))\) that are analytically nondegenerate. The main result of the paper is the following theorem: If \(\text{rank} \varPsi=\dim M\) and \(d>(s+1)!\{(s+1)!-2\},\) then the number of mappings in \(\mathcal F^\ast(p;(\mathbb C^m,E),(M,D))\) is bounded by a constant depending only on \(D\). Remark. For divisors \(E_1\) and \(E_2\) on \(\mathbb C^m\) we write \(E_1\equiv E_2\pmod p\) if there exists a divisor \(E'\) on \(\mathbb C^m\) such that \(E_1-E_2=pE'\). in special case of \(p=0, E_1\equiv E_2\pmod 0\) iff \(E_1=E_2.\) Let \(E\) be a nonzeroeffective divisor on \(\mathbb C^m.\) The finiteness problem for meromorphic mappings under the condition on the preimages of divisors was first studied by H. Cartan and R. Nevanlinna and they obtained a finiteness theorem for mero…
Mathematics Subject Classification ID: 32H35
zbMATH DE Number: 1216674
zbMATH Keywords: meromorphic mappings
zbMATH Keywords: finiteness theorems
zbMATH Keywords: divisors
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: inverse images
zbMATH Keywords: Borel's identity
zbMATH Keywords: compact complex manifolds
zbMATH Keywords: analytic dependence
zbMATH Keywords: uniqueness problems
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1216674