The Geometry of Weakly Self-dual Kähler Surfaces

We study Kähler surfaces with harmonic anti-self-dual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kähler metrics, including Kähler–Einstein metrics and conformally Einstein Kähler metrics. We also extend some of our results to almost Kähler 4-manifolds, providing new examples of Ricci-flat almost Kähler metrics which are not Kähler.