arXiv:math/0105112v1 [math.DG] 14 May 2001 2000]Primary 53C55; Secondary 14M25, 53D20. K ¨ AHLER 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” action-angle 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 K¨ahler metrics and generalizes an analogous result for toric manifolds. A simple explicit description of interesting families of extremal K¨ahler met- rics, 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. 1. Introduction The space of K¨ahler metrics on a K¨ahler manifold (or orbifold) can be described in two equivalent ways, reflecting the fact that a K¨ahler manifold is both a complex and a symplectic manifold. From the complex point of view, one starts with a fixed complex manifold (M,J 0 ) and K¨ahler class Ω ∈ H 1,1 J0 ∩ H 2 (M, R), and considers the space S (J 0 , Ω) of all symplectic forms ω on M that are compatible with J 0 and represent the class Ω. Any such form ω ∈S (J 0 , Ω) gives rise to a K¨ahler metric 〈·, ·〉 ≡ ω(·,J 0 ·). The symplectic point of view arises naturally from the observation that any two forms ω 0 ,ω 1 ∈S (J 0 , Ω) define equivalent symplectic structures on M . In fact, the family ω t = ω 0 + t(ω 1 − ω 0 ), for t ∈ [0, 1], is an isotopy of symplectic forms in the same cohomology class Ω, and so Moser’s theorem [M] gives a family of diffeomorphisms ϕ t : M → M,t ∈ [0, 1], such that ϕ ∗ t (ω t )= ω 0 . In particular, the K¨ahler manifold (M,J 0 ,ω 1 ) is K¨ahler isomorphic to (M,J 1 ,ω 0 ), where J 1 = (ϕ 1 ) −1 ∗ ◦ J 0 ◦ (ϕ 1 ) ∗ . This means that one can also describe the space of K¨ahler metrics starting with a fixed symplectic manifold (M,ω 0 ) and considering the space J (ω 0 , [J 0 ]) of all complex structures J on M that are compatible with ω 0 and belong to some diffeo- morphism class [J 0 ], determined by a particular compatible complex structure J 0 . Any such J ∈J (ω 0 , [J 0 ]) gives rise to a K¨ahler metric 〈·, ·〉 ≡ ω 0 (·,J ·). Date : November 10, 2018. 1991 Mathematics Subject Classification. [. Key words and phrases. Symplectic toric orbifolds, K¨ahler metrics, action-angle coordinates, extremal metrics, self-dual Einstein metrics. Partially supported by FCT (Portugal) through program POCTI and grant POCTI/1999/MAT/33081. The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by the European Human Potential Programme. 1