Weighted cscK metric (I): a priori estimates

Abstract

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

There was a mistake in Lemma 5.5 and Lemma 5.6 where an important term in the estimates was wrongly neglected. We corrected them. The $C^2$ estimate now holds under the assumption that the weight $\mathrm{v}$ is log-concave