Toric Kähler–Einstein Metrics and Convex Compact Polytopes

Abstract

We show that any compact convex simple lattice polytope is the moment polytope of a Kahler–Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang–Zhu Theorem (Wang and Zhu in Adv Math 188:47–103, 2004) giving the existence of a Kahler–Ricci soliton on any toric monotone manifold on any compact convex simple labeled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope Open image in new window admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condition (with respect to these normals) and the Kahler–Einstein equation on Open image in new window. We interpret our result in terms of existence of singular Kahler–Einstein metrics on toric manifolds.