Algebraic dimension of twistor spaces and scalar curvature of anti-self- dual metrics
type: Item
instance of: scholarly article
title: Algebraic dimension of twistor spaces and scalar curvature of anti-self- dual metrics
zbMATH Open document ID: 0747.32021
DOI: 10.1007/BF01445194
author: Massimiliano Pontecorvo
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164853
review text: Let \(t:Z\to M\) be the twistor fibration of an anti-self-dual 4-manifold \((M,g)\). We prove that the algebraic dimension \(a(Z)\) of \(Z\) is at most 1 when the conformal class \([g]\) contains a metric of zero scalar curvature and that equality holds exactly when this metric is Ricci-flat. We also show that if \(a(Z)\) is at least 2 then \([g]\) contains a metric of positive scalar curvature. These are generalizations of theorems of \textit{Y. S. Poon} [Proc. Am. Math. Soc. 111, No. 2, 331-338 (1991; Zbl 0682.53067) and Math. Ann. 282, No. 4, 621-627 (1988; Zbl 0665.32014)] and together with a result of Gauduchon-Ville [\textit{P. Gauduchon}, `Les structures holomorphes du fibré tangent vertical sur l'espace des twisteur d'une variété conforme autoduale'. Preprint (1989)] give a fairly complete account on the relation between \(a(Z)\) and the sign of the scalar curvature of the conformal class \([g]\).
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 32A20
Mathematics Subject Classification ID: 32L25
zbMATH DE Number: 59952
zbMATH Keywords: twistor space
zbMATH Keywords: algebraic dimension
zbMATH Keywords: scalar curvature
MaRDI profile type: Publication
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Wikidata QID: Q125339988
description: scientific article; zbMATH DE number 59952