Coupled K\"ahler-Ricci solitons on toric Fano manifolds

TL;DR

This paper establishes a precise criterion based on polytopes' barycenters for the existence of coupled K"ahler-Einstein metrics and soliton solutions on toric Fano manifolds, confirming a conjecture in this setting.

Contribution

It provides necessary and sufficient conditions for coupled K"ahler-Einstein metrics and soliton solutions on toric Fano manifolds, confirming a version of the Yau-Tian-Donaldson conjecture.

Findings

Characterization of coupled K"ahler-Einstein metrics via barycenter conditions

Necessary and sufficient conditions for torus-invariant soliton solutions

Identification of potential limits of a geometric flow generalizing K"ahler-Ricci flow

Abstract

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled K\"ahler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. 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. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the K\"ahler-Ricci flow.