Vol. 12, No. 8, 2019

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Coupled Kähler–Ricci solitons on toric Fano manifolds

Jakob Hultgren

Vol. 12 (2019), No. 8, 2067–2094
Abstract

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler–Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau–Tian–Donaldson conjecture and as a corollary we obtain an example of a coupled Kähler–Einstein metric on a manifold which does not admit Kähler–Einstein metrics. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton-type equations on toric Fano manifolds.

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Keywords
coupled Kähler–Einstein metrics, Kähler–Einstein metrics, Monge–Ampère equations, Kähler manifolds
Mathematical Subject Classification 2010
Primary: 32Q15, 32Q20, 32Q26, 53C25
Milestones
Received: 9 April 2018
Accepted: 18 October 2018
Published: 28 October 2019
Authors
Jakob Hultgren
Matematisk Institutt
Universitetet i Oslo
Oslo
Norway