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 e#ective 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

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

Collections: Mathematics