Weighted cscK metric (I): a priori estimates

TL;DR

This paper establishes a priori estimates for weighted cscK metrics on compact K"ahler manifolds, extending classical results to a more general weighted setting and providing foundational analytical tools for their study.

Contribution

It introduces a priori $C^{k}$-estimates for weighted cscK metrics, broadening the understanding of their existence and regularity beyond classical cscK metrics.

Findings

Established $C^{k}$-estimates for weighted cscK metrics

Extended Chen and Cheng's classical results to weighted settings

Provided analytical groundwork for future existence proofs

Abstract

Let $X$ be a compact K\"ahler manifold. In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.