Coupled Kähler–Ricci solitons on toric Fano manifolds

Hultgren, Jakob

doi:10.2140/apde.2019.12.2067

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Coupled Kähler–Ricci solitons on toric Fano manifolds

Jakob Hultgren

Vol. 12 (2019), No. 8, 2067–2094

DOI: 10.2140/apde.2019.12.2067

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.

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

Vol. 12, No. 8, 2019