Peng Wu [P. Wu, “Einstein four-manifolds with self-dual Weyl tensor of nonnegative determinant”, Int. Math. Res. Not. (2021), no. 2, 1043–1054] recently announced a beautiful characterization of conformally Kähler, Einstein metrics of positive scalar curvature on compact oriented $4$-manifolds via the condition $\operatorname{det}(W^{+}) \gt 0$. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in [C. LeBrun, “Einstein metrics, harmonic forms, and symplectic four-manifolds,” Ann. Global Anal. Geom. 48 (2015), no. 1, 75–85].

中文翻译：

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爱因斯坦流形，自双Weyl曲率和保形Kähler几何

彭武[P. Wu，“具有非负行列式自对偶Weyl张量的爱因斯坦四流形”，《诠释学》。数学。Res。不是。（2021年），没有。[2，1043–1054]最近宣布了通过条件$ \ operatorname {det}（W ^ {+}）\ gt 0 $对紧致的$ 4 $流形上的正标量曲率的爱因斯坦共形Kähler，Einstein度量的优美刻画。在本说明中，我们通过提供与他的结果完全不同的证据来支持他的主张。然后，我们将基于先前在[C. LeBrun，“爱因斯坦度量，调和形式和辛四流形”，安。全球肛门。几何 48（2015），第。1，75–85]。