Toric Kahler-Einstein Metrics and Convex Compact Polytopes

摘要

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 P subset of R-n 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 P x R-n. We interpret our result in terms of existence of singular Kahler-Einstein metrics on toric manifolds.