TL;DR
This paper classifies compact 4-dimensional Einstein manifolds with Hermitian metrics, showing they are either Kähler-Einstein or belong to two specific exceptional cases, extending understanding of Einstein geometry in four dimensions.
Contribution
It provides a classification of Hermitian Einstein 4-manifolds, identifying two unique non-Kähler examples beyond the Kähler-Einstein case.
Findings
Hermitian Einstein 4-manifolds are either Kähler-Einstein or one of two exceptional metrics.
The Page metric on CP2 # (-CP2) is characterized as an exception.
The Chen-LeBrun-Weber metric on CP2 # 2 (-CP2) is identified as another exception.
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 Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2 # 2 (-CP2) constructed in Chen-LeBrun-Weber.
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