Toric Kähler–Einstein Metrics and Convex Compact Polytopes

Abstract

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem [41] giving the existence of a Kähler-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 ⊂ 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 Kähler-Einstein equation on P × R n . We interpret our result in terms of existence of singular Kähler-Einstein metrics on toric manifolds.

A Note on Extremal Toric Almost Kähler Metrics

“…They are many ways to construct the corresponding (compact) toric symplectic orbifold (M, ω, T := t/Λ) from the data (P, n, Λ). We recall only the one we will use which, as far as we know, has been developped in [20,16,30].…”

“…where T = t/Λ we get a way to glue equivariantly the (uniformizing) chart M p over P × T seen as a toric symplectic manifold with momentum map x being the projection on the first factor, see [30] for more details.…”

“…To get around the lack of compacity of P , Donaldson argue that the system of charts associated to the vertices, see §2.1.2, provides the kind of compactification needed. This idea is developped with details in [30].…”

“…Indeed, in this case U would contain a basis {σ s } s=1,...,d ⊂ U and anyσ ∈ M(P ) is such σ = d s=1 a s σ s with a s ∈ R. Picking any solution H s ∈ W(σ s ) we have d s=1 a s H s ∈ W(σ). According to [30] for each polytope there exists σ KE ∈ M(P ), unique up to dilatation, and a symplectic potential u KE ∈ S(P, n σKE ) such that the metric g uKE is Kähler-Einstein on P × t with respect to the natural symplectic structure on t * × t. The assertion (i) is that if there exists an extremal toric almost Kähler metric compatible with ω then there exists an extremal toric Kähler metric and assertion (ii) is that if (P, σ) is uniformly K-stable then there exists an extremal toric almost Kähler metric compatible with ω. This is Corollary 1.4.…”

“…[30] Let (M, ω, T ) be a toric compact symplectic manifold with labelled moment polytope (P, m, Λ) and momentum map x : M → t * . For any labelling n of P , any potential u ∈ S(P, n) provides a Kähler metric g u , defined via (9), smooth and compatible with ω onM = x −1 (P ) and with cone angle singularity 2π( n s / m s ) transverse to the divisor x −1 (F s ).…”