Weighted cscK metrics on K\"ahler varieties

TL;DR

This paper extends the theory of weighted constant scalar curvature K"ahler metrics to mildly singular varieties, establishing existence results under certain conditions and introducing a new construction method that bypasses traditional gluing techniques.

Contribution

It generalizes the existence of singular weighted cscK metrics to mildly singular K"ahler varieties using a novel approach that does not rely on detailed metric behavior near singularities.

Findings

Existence of singular weighted cscK metrics under coercivity conditions.

Extension of Chen-Cheng and He’s results to singular settings.

A new construction method inspired by Arezzo-Pacard that avoids detailed local analysis.

Abstract

We study the weighted constant scalar curvature K\"ahler equations on mildly singular K\"ahler 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.