Complex structures on Riemannian four-manifolds
type: Item
instance of: scholarly article
title: Complex structures on Riemannian four-manifolds
zbMATH Open document ID: 0893.53026
author: Massimiliano Pontecorvo
published in: Mathematische Annalen
publication date: 1997-09-09T00:00:00Z
review text: The purpose of this work is to investigate those connected oriented Riemannian 4-manifolds admitting several compatible complex structures which are independent. It is shown that \((M^4,g)\) is necessarily anti-self-dual if there exists three compatible complex structures \(J_1\), \(J_2\), \(J_3\) such that for each \(i\neq j\) there is a point \(p\in M\), where \(J_i(p)\neq\pm J_j(p)\). Moreover, if \(M\) is a compact anti-selfdual manifold admitting two compatible complex structures \(J_1\neq\pm J_2\), then, up to a conformal rescaling of the metric, it is a flat torus, a Ricci-flat Kähler K3 surface, a conformally flat Hopf surface or it is diffeomorphic to \((S^1\times S^3)\# n\overline{\mathbb{C} P^2}\), \(n\geq 1\). Also, applications to the existence of orientation-preserving isometries, Killing vector fields and hyper-Hermitian structures are given.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32L25
zbMATH DE Number: 1058824
zbMATH Keywords: compatible independent complex structures
zbMATH Keywords: anti-selfduality
zbMATH Keywords: isometries
zbMATH Keywords: Killing vector fields
zbMATH Keywords: hyper-Hermitian structures
Wikidata QID: Q115388975
reviewed by: Eduardo García-Río
MaRDI profile type: Publication
full work available at URL: s002080050108
OpenAlex ID: W2063772150
DOI: 10.1007/S002080050108
description: scientific article; zbMATH DE number 1058824