Summary: K¨AHLER METRICS ON TORIC ORBIFOLDS
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.
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