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.
"Coupled Kähler–Ricci solitons on toric Fano manifolds." Anal. PDE 12 (8) 2067 - 2094, 2019. https://doi.org/10.2140/apde.2019.12.2067