On relations between Mabuchi’s generalized Kähler-Einstein metrics and various canonical Kähler metrics

Abstract

In this short note, we prove that a generalized Kähler-Einstein metric g g on a Fano manifold is actually a Kähler-Einstein metric if and only if one of the following conditions is satisfied: (i) g g is a Kähler-Ricci soliton; (ii) g g is an extremal Kähler metric.