On the holomorphic invariants for generalized Kähler-Einstein metrics

Abstract

In [9], Mabuchi extended the notion of Kähler-Einstein metrics to the case of Fano manifolds with novanishing Futaki invariant. We call them generalized Kähler-Einstein metrics. He defined the holomorphic invariant α_M in terms of the extremal Kähler vector field, which is the obstruction for the existence of generalized Kähler-Einstein metrics. The purpose of this short paper is to show that the above obstruction is actually equivalent to the vanishing of the holomorphic invariant of Futaki's type defined by Futaki [4] (see also [8]). As its corollary, we can show that $\mathbf{CP}^2\sharp \overline{\mathbf{CP}^2}$ admits generalized Kähler-Einstein metrics by the method using multiplier ideal sheaves in [6].