Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry

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].

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This work was supported in part by NSF grant DMS-1906267, and was undertaken at the conclusion of the author’s sabbatical year as a Simons Fellow in Mathematics.

Received 5 August 2019

Accepted 25 May 2020

Published 24 May 2022