Toric Kähler-Einstein metrics and convex compact polytopes

Details Toric Kähler-Einstein metrics and convex compact polytopes Toric Kähler-Einstein metrics and convex compact polytopes

2011-12-14

arxiv.org/abs/1112.3239v1

Mathematics

Differential Geometry

21 pages

Scientific paper

We show that any compact convex simple lattice polytope is the moment polytope of a K\"ahler-Einstein orbifold, unique up to orbifold covering and homothety. Using the symplectic approach of Donaldson [12], we extend the Wang-Zhu Theorem [35] giving the existence of a K\"ahler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ 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 K\"ahler-Einstein equation on $P\times \bR^n$.

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