Deformation for coupled Kähler–Einstein metrics

Abstract

The notion of coupled Kähler–Einstein metrics was introduced recently by Hultgren–Witt Nyström. In this paper we discuss deformation of a coupled Kähler–Einstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler–Einstein metric to be deformed to another coupled Kähler–Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher–Einstein metric on a Fano manifold when the complex structure varies.