Journal of Differential Geometry

On Einstein, Hermitian 4-manifolds

Claude LeBrun

Abstract

Let $(M, h)$ be a compact 4-dimensional Einstein manifold, and suppose that $h$ is Hermitian with respect to some complex structure $J$ on $M$. Then either $(M, J, h)$ is Kähler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on $\mathbb{CP}_2\#\overline{\mathbb{CP}}_2$, or the Einstein metric on $\mathbb{CP}_2\#2\overline{\mathbb{CP}}_2$ discovered in "On conformally Kähler, Einstein manifolds," J. Amer. Math. Soc. 21 (2008), pp. 1137–1168.

Article information

Source
J. Differential Geom., Volume 90, Number 2 (2012), 277-302.

Dates
First available in Project Euclid: 24 April 2012

https://projecteuclid.org/euclid.jdg/1335230848

Digital Object Identifier
doi:10.4310/jdg/1335230848

Mathematical Reviews number (MathSciNet)
MR2899877

Zentralblatt MATH identifier
1254.53076

Citation

LeBrun, Claude. On Einstein, Hermitian 4-manifolds. J. Differential Geom. 90 (2012), no. 2, 277--302. doi:10.4310/jdg/1335230848. https://projecteuclid.org/euclid.jdg/1335230848