 
Summary: K˜ahler Geometry of Toric Varieties and Extremal
Metrics
Miguel Abreu #
Institute for Advanced Study
final version
March 4, 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 # # R n . Recently Guillemin gave
an explicit combinatorial way of constructing ``toric'' K˜ahler metrics on
X, using only data on #. In this paper, di#erential 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 a#ne function on
# # R n . A construction, due to Calabi, of a 1parameter family of ex
tremal K˜ahler metrics of nonconstant scalar curvature on CP 2 #CP 2 is
recast very simply and explicitly using Guillemin's approach. Finally, we
