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Toric Kähler–Einstein Metrics and Convex Compact Polytopes

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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 (Wang and Zhu in Adv Math 188:47–103, 2004) giving the existence of a Kähler–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 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 . We interpret our result in terms of existence of singular Kähler–Einstein metrics on toric manifolds.

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Notes

  1. Our convention is that \(P\) is open and we denote \(\overline{P}=\{p\in \mathfrak {t}^*\,|\,L_k(p)\ge 0\}\). \(\overline{P}\) is compact.

  2. We denote \(P\) the interior of the polytope and \(\overline{P}\) its closure. In this text, polytopes are always assumed to be convex and simple with compact closure.

  3. The argument is stated for \(n=2\) but holds in general.

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Acknowledgments

The author is grateful to Professor Vestislav Apostolov for bringing her attention to this problem and for his numerous advice. This project ended during a stay at the MIT, the author thanks Professor Victor Guillemin for his interest in this problem. She also thanks Yanir Rubinstein for his explanations of some subtleties concerning singular Kähler metrics. Finally, she thanks the referees for pointing to her a mistake in a previous version of this work and giving her ideas to complete a proof.

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  1. I.M.T., Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    Eveline Legendre

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  1. Eveline Legendre
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Legendre, E. Toric Kähler–Einstein Metrics and Convex Compact Polytopes. J Geom Anal 26, 399–427 (2016). https://doi.org/10.1007/s12220-015-9556-z

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