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 $t\in (0, δ)$ there exists a unique cscK metric $g_t$ in $K_X+ t γ$. In this paper, we prove that $\{ (X, g_t) \}_{ t\in (0, δ)}$ have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.

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