Kahler geometry of toric varieties and extremal metrics

TL;DR

This paper explores the differential geometry of toric Kahler metrics constructed via Guillemin's combinatorial method, providing formulas for scalar curvature, extremality conditions, and a new perspective on Calabi's extremal metrics.

Contribution

It introduces explicit combinatorial formulas for scalar curvature and extremal conditions of toric Kahler metrics, simplifying Calabi's construction and revealing new relations with convex polytopes.

Findings

Derived a combinatorial formula for scalar curvature.

Established the Euler-Lagrange condition for extremal metrics.

Presented a new combinatorial formula relating scalar curvature and polytope geometry.

Abstract

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. 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 is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.