 
Summary: 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"
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 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 selfdual Einstein metrics connecting the well known
EguchiHanson and TaubNUT 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
