CscK metrics near the canonical class

Abstract

Let 𝑋 be a Kähler manifold with semiample canonical bundle

. It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models, Proc. Amer. Math. Soc. 147 (2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists

δ > 0 \delta>0

such that, for all

t ∈ ( 0 , δ ) t\in(0,\delta)

, there exists a unique cscK metric

g t g_{t}

in

K X + t ⁢ γ K_{X}+t\gamma

. In this paper, we prove that

{ ( X , g t ) } t ∈ ( 0 , δ ) \{(X,g_{t})\}_{t\in(0,\delta)}

have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense.