Weighted cscK metrics on Kähler varieties
Chung-Ming Pan, Tat Dat Tô
TL;DR
This work develops a comprehensive analytic framework for weighted constant scalar curvature Kähler (cscK) metrics on mildly singular Kähler varieties. It extends the coercivity–existence paradigm of Chen–Cheng and He to the singular weighted setting by solving a family of perturbed weighted cscK equations on a resolution and passing to a singular limit, while establishing uniform $L^\infty$, Laplacian, and higher-order estimates. A central result shows that if the weighted Mabuchi functional $\mathbf{M}_{v,w}$ is $T_{\mathbb{C}}$-coercive, then a singular $(v,w)$-cscK metric exists in the given Kähler class and minimizes the functional. The paper also provides a robust method to construct singular cscK metrics via Arezzo–Pacard-type blowups, blended with smoothing techniques, yielding numerous new examples on singular spaces and illustrating the weighted framework's versatility for extremal metrics and beyond.
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.