 
Summary: K¨ahler Geometry of Toric Varieties and Extremal
Metrics
Miguel Abreu
Institute for Advanced Study
final version
October 28, 1999
1991 Mathematics Subject Classification: Primary 53C55, Secondary 14M25
53C25 58F05.
Abstract
A (symplectic) toric variety X, of real dimension 2n, is completely
determined by its moment polytope Rn
. Recently Guillemin gave
an explicit combinatorial way of constructing "toric" K¨ahler metrics on
X, using only data on . In this paper, differential geometric properties
of these metrics are investigated using Guillemin's construction. In par
ticular, a nice combinatorial formula for the scalar curvature R is given,
and the EulerLagrange condition for such "toric" metrics being extremal
(in the sense of Calabi) is proven to be R being an affine function on
Rn
. A construction, due to Calabi, of a 1parameter family of ex
