## Kodai Mathematical Journal

### On the holomorphic invariants for generalized Kähler-Einstein metrics

Yuji Sano

#### 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 αM 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].

#### Article information

Source
Kodai Math. J., Volume 31, Number 3 (2008), 431-440.

Dates
First available in Project Euclid: 6 November 2008

https://projecteuclid.org/euclid.kmj/1225980446

Digital Object Identifier
doi:10.2996/kmj/1225980446

Mathematical Reviews number (MathSciNet)
MR2475279

Zentralblatt MATH identifier
1167.53062

#### Citation

Sano, Yuji. On the holomorphic invariants for generalized Kähler-Einstein metrics. Kodai Math. J. 31 (2008), no. 3, 431--440. doi:10.2996/kmj/1225980446. https://projecteuclid.org/euclid.kmj/1225980446