Harmonic mappings and disc bundles over compact Kähler manifolds
type: Item
instance of: scholarly article
title: Harmonic mappings and disc bundles over compact Kähler manifolds
zbMATH Open document ID: 0601.32023
author: K. Diederich
author: Takeo Ohsawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1985-01-01T00:00:00Z
review text: Theorem 1. Let X be a compact Kähler manifold and \(\Omega\to^{\pi}X\) a locally trivial holomorphic disc bundle. Then \(\Omega\) is weakly 1- complete. The proof uses the harmonic sections with respect to the Kähler metric \(ds^ 2_ X\) on X and the Kähler metric \(ds^ 2\) induced on \(\Omega\) by \(ds^ 2_ X\) and the Poincaré metric \(ds^ 2_ h\) on \(\Delta\). The authors provide the following existence theorem for harmonic sections. Theorem 2. Let \(\Omega\to^{\pi}X\) be a locally trivial holomorphic disc bundle over the compact Kähler manifold X. Suppose that the corresponding \(P^ 1\)-bundle \({\hat \Omega}\to^{{\hat \pi}}X\) does not allow a flat section in \(\partial \Omega\). Then there exists a harmonic section \(s: X\to \Omega.\) Then Theorem 1 is generalized to Theorem 3. Let X be a compact complex manifold which is bimeromorphically equivalent to a compact Kähler manifold. Then any locally trivial holomorphic disc bundle \(\Omega\to^{\pi}X\) is weakly 1-complete. An extendibility result for harmonic maps is also given.
Mathematics Subject Classification ID: 32F99
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 58C05
zbMATH DE Number: 3969061
zbMATH Keywords: compact Kähler manifold
zbMATH Keywords: holomorphic disc bundle
zbMATH Keywords: harmonic section
zbMATH Keywords: bimeromorphically equivalent
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178932
OpenAlex ID: W2024891652
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DOI: 10.2977/PRIMS/1195178932
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description: scientific article; zbMATH DE number 3969061