Kahler geometry of toric varieties and extremal metrics

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope a Delta subset of R-n. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on Delta. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler-Lagrange condition for such "toric" metrics being extremal tin the sense of Calabi) is proven to be R being an affine function on Delta subset of R-n. A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on CP2 #<(CP)over bar>(2) is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Delta subset of R-n that follows from the wellknown relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class.