Weighted cscK metrics on Kähler varieties

Abstract We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen–Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo–Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.