Song–Tian's Gromov–Hausdorff convergence conjecture for Kähler–Ricci flow

Let $(X^n,\omega_0)$ be a compact Kähler manifold with semiample canonical bundle $K_X$. Let $\omega(t)$ solve the normalized Kähler–Ricci flow

$$\frac{\partial}{\partial t}\omega(t)=-\operatorname{Ric}(\omega(t))-\omega(t),\qquad \omega(0)=\omega_0.$$

Let $f:X\to Y$ be the canonical-model morphism, let $Y^\circ$ be the complement of the singularities and discriminant locus, and let $\omega_{\rm can}$ be the twisted Kähler–Einstein metric on $Y^\circ$. Song–Tian's Gromov–Hausdorff convergence conjecture. As $t\to\infty$, $(X,\omega(t))$ converges in the Gromov–Hausdorff topology to the metric completion of $(Y^\circ,\omega_{\rm can})$, which is a compact metric space homeomorphic to the canonical model $Y$. The conjecture belongs to the analytic minimal model program and concerns the global, rather than merely local smooth, limiting behavior of collapsing Kähler–Ricci flows. The result is established in the paper from which this statement is taken.