On conformally Kähler, Einstein manifolds

Chen, Xiuxiong; LeBrun, Claude; Weber, Brian

doi:10.1090/S0894-0347-08-00594-8

On conformally Kähler, Einstein manifolds
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by Xiuxiong Chen, Claude LeBrun and Brian Weber;

J. Amer. Math. Soc. 21 (2008), 1137-1168

DOI: https://doi.org/10.1090/S0894-0347-08-00594-8

Published electronically: January 28, 2008

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Abstract:

We prove that any compact complex surface with $c_1>0$ admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up $\mathbb {CP}_2\# 2\overline {\mathbb {CP}_2}$ of the complex projective plane at two distinct points.

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