Twistor spaces, Einstein metrics and isomonodromic deformations
type: Item
instance of: scholarly article
title: Twistor spaces, Einstein metrics and isomonodromic deformations
zbMATH Open document ID: 0861.53049
published in: Journal of Differential Geometry
publication date: 1997-05-13T00:00:00Z
review text: The importance of twistor theory results from its ability to translate differential geometric questions into equivalent problems in algebraic geometry. For example, an anti-self-dual conformal structure on a four-manifold \(M\) can be encoded as the holomorphic structure of a complex three-manifold, its twistor space. In this paper, anti-self-dual manifolds \(M\) are studied, which carry an action of the Lie group SU(2), preserving the conformal structure and having three-dimensional orbits. The natural lifting of this action on the twistor space \(Z\) defines a homomorphism of holomorphic vector bundles \(\alpha:{\mathcal O}_Z\otimes {\mathfrak g}^c\to TZ\), whose rank is at least two. If the image sheaf of \(\alpha\) has rank two, then he shows the conformal structure on \(M\) to be either conformally flat, or the Eguchi-Hanson metric, or a member of the family described by \textit{V. A. Belinskiĭ, G. W. Gibbons, D. N. Page} and \textit{C. N. Pope} [Phys. Lett. B 76, No. 4, 433-435 (1978)]. The bulk of this paper is concerned with the case where \(\alpha\) has image of rank three. In this case, \(\alpha\) fails to be an isomorphism over an anticanonical divisor \(Y\subset Z\). The case where \(Y=2D\) is a multiple of a half-anticanonical divisor leads to Painlevé's third equation and is studied in a paper of \textit{H. Pedersen} and \textit{Y. S. Poon} [Classical Quantum Gravity 7, No. 10, 1707-1719 (1990; Zbl 0711.53039)]. The author studies in full detail the case where the divisor \(Y\) intersects a generic twistor line at four distinct points. On a generic twistor line, outside these four intersection points the morphism \(\alpha\) defines a flat connection whose holonomy representation does not depend on the twistor line. Such a family of connections is called an isomonodromic deformation. The residues of these connections on the four points of intersection define four trace-free \(2\times 2\) matrices \(A_i\), fulfilling the Schlesinger equation [see \textit…
reviewed by: Bernd Kreußler
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 32V05
zbMATH DE Number: 844214
zbMATH Keywords: scalar-flat Kähler metric
zbMATH Keywords: meromorphic connection
zbMATH Keywords: twistor space
zbMATH Keywords: anti-self-dual manifolds
zbMATH Keywords: conformal structure
zbMATH Keywords: Einstein metric
zbMATH Keywords: monodromy group
zbMATH Keywords: Painlevé VI
zbMATH Keywords: theta functions
author: Nigel Hitchin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214457032
OpenAlex ID: W1503703259
Wikidata QID: Q115178007
DOI: 10.4310/JDG/1214457032
description: scientific article; zbMATH DE number 844214