KAHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS

A (symplectic) toric variety X, of real dimension 2n, is completely de termined by its moment polytope a Delta subset of R-n. Recently Guille min gave an explicit combinatorial way of constructing ''toric'' Kahle r 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 s calar 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 integr al of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suita ble power of the Kahler class.