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The Geometry of Weakly Self-dual Kähler Surfaces

Published online by Cambridge University Press:  04 December 2007

V. Apostolov ,
D. M. J. Calderbank  and
P. Gauduchon
V. Apostolov
Affiliation:
Département de Mathématiques, UQAM, CP 8888, Succ. Centre-ville, Montréal, Québec H3C 3P8, Canada. e-mail: apostolo@math.uqam.ca
D. M. J. Calderbank
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, King's Building, Mayfield's Road, Edinburgh EH9 3JZ, Scotland. e-mail: davidmjc@maths.ed.ac.uk
P. Gauduchon
Affiliation:
Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France. e-mail: pg@math.polytechnique.fr
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Abstract

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We study Kähler surfaces with harmonic anti-self-dual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kähler metrics, including Kähler–Einstein metrics and conformally Einstein Kähler metrics. We also extend some of our results to almost Kähler 4-manifolds, providing new examples of Ricci-flat almost Kähler metrics which are not Kähler.

Keywords

almost-Kähler 4-manifoldsCalabi extremal metricsKähler surfacesweak self-duality
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 3 , February 2003 , pp. 279 - 322
Copyright
© 2003 Kluwer Academic Publishers