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

Let $(X^n,\omega_0)$ be a compact Kähler manifold with $K_X$ semiample and numerical dimension $0<m<n$. Let $\omega(t)$ solve the normalized Kähler-Ricci flow, and let $Y$, $Y^\circ$, and $\omega_{\rm can}$ be as in the associated canonical fibration.

Song–Tian's Gromov–Hausdorff conjecture.

(a) There is $C>0$ such that

$$\operatorname{diam}(X,\omega(t))\leqslant C$$

for all $t\geqslant0$.

(b) As $t\to+\infty$, $(X,\omega(t))$ converges in the Gromov–Hausdorff topology to $(Z,d)$, where $(Z,d)$ is the metric completion of $(Y^\circ,\omega_{\rm can})$.

(c) $Z$ is homeomorphic to $Y$.

The conjecture describes the global metric limit of the collapsing Kähler-Ricci flow, beyond the known local estimates away from singular fibers. Some cases are known, but the general assertions were presented as conjectural in the cited discussion.