Convergence of cscK metrics on smooth minimal models of general type

We consider constant scalar curvature Kähler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed Kähler metric. We show that sequences of such metrics converge smoothly on compact subsets away from a subvariety to the singular Kähler–Einstein metric in the canonical class. This confirms partially a conjecture of Jian–Shi–Song about the convergence behavior of such sequences.