A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes
type: Item
instance of: scholarly article
title: A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes
zbMATH Open document ID: 0979.53082
author: Nobuhiro Honda
author: Mitsuhiro Itoh
published in: Journal of the Mathematical Society of Japan
publication date: 2000-08-27T00:00:00Z
review text: We show that there exist on \(4\mathbb{C}\mathbb{P}^2\), the connected sum of four complex projective planes, self-dual metrics with the following properties: (i) the sign of the scalar curvature is positive, (ii) the identity component of the isometry group is \(U(1)\), (iii) the metrics are not conformally isometric to the self-dual metrics constructed by \textit{C. Le Brun} [J. Differ. Geom. 34, 223-253 (1991; Zbl 0725.53067)]. These are the first examples of self-dual metrics with non-semi free \(U(1)\)-isometries on simply connected manifolds. Our proof is based on the twistor theory: we use an equivariant orbifold version of the construction of \textit{S. Donaldson} and \textit{R. Friedman} [Nonlinearity 2, 197-239 (1989; Zbl 0671.53029)]. We also give a rough description of the structure of the algebraic reduction of the corresponding twistor spaces.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C28
zbMATH DE Number: 1439295
zbMATH Keywords: elliptic fibration
zbMATH Keywords: connected sum
zbMATH Keywords: complex projective planes
zbMATH Keywords: scalar curvature
zbMATH Keywords: isometry group
zbMATH Keywords: self-dual metrics
zbMATH Keywords: twistor theory
zbMATH Keywords: twistor spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05210139
OpenAlex ID: W2092721905
Wikidata QID: Q126282255
DOI: 10.2969/JMSJ/05210139
description: scientific article; zbMATH DE number 1439295