CscK metrics near the canonical class

Abstract

Let $X$ be a K\"ahler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any K\"ahler class $\gamma$, there exists $\delta>0$ such that for all $t\in (0, \delta)$ there exists a unique cscK metric $g_t$ in $K_X+ t \gamma $. In this paper, we prove that $\{ (X, g_t) \}_{ t\in (0, \delta)} $ have uniformly bounded K\"ahler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.