Journal of Differential Geometry

Kähler Metrics on Toric Orbifolds

Miguel Abreu

Abstract

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" actionangle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kähler metrics and generalizes an analogous result for toric manifolds.

A simple explicit description of interesting families of extremal Kähler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.

Article information

Source
J. Differential Geom., Volume 58, Number 1 (2001), 151-187.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090348285

Digital Object Identifier
doi:10.4310/jdg/1090348285

Mathematical Reviews number (MathSciNet)
MR1895351

Zentralblatt MATH identifier
1035.53055

Citation

Abreu, Miguel. Kähler Metrics on Toric Orbifolds. J. Differential Geom. 58 (2001), no. 1, 151--187. doi:10.4310/jdg/1090348285. https://projecteuclid.org/euclid.jdg/1090348285


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