Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds
type: Item
instance of: scholarly article
title: Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds
zbMATH Open document ID: 0648.53037
DOI: 10.1007/BF01161998
author: Chi-Keung Cheung
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174038
review text: It is well-known that if a compact complex manifold admits a Hermitian metric with negative holomorphic sectional curvature, then it is hyperbolic. The converse is still open. The purpose of this note is to show that for some types of compact hyperbolic manifolds, we can construct a Hermitian metric on them with everywhere negative holomorphic sectional curvature. In particular, Kodaira's surfaces of general type and the example of \textit{H. Grauert} and \textit{H. Reckziegel} [Math. Z. 89, 108-125 (1965; Zbl 0135.125)] are included as special cases, while the local result of the latter is extended to a global one. In the case of Kodaira surfaces, we will show that they can in fact be given a Kähler metric with negative holomorphic sectional curvature.
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4058356
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: Hermitian metric
zbMATH Keywords: negative holomorphic sectional curvature
zbMATH Keywords: Kodaira's surfaces
zbMATH Keywords: Kähler metric
MaRDI profile type: Publication
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OpenAlex ID: W2011330513
description: scientific article; zbMATH DE number 4058356