CscK metrics near the canonical class

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