Hamilton–Tian conjecture for the Kähler–Ricci flow on Fano manifolds

Let $X$ be a Fano manifold and let $\omega(t)$ be a global solution of the normalized Kähler–Ricci flow

$$\frac{\partial \omega(t)}{\partial t}=-\operatorname{Ric}(\omega(t))+\omega(t).$$

A $\mathbb{Q}$-Fano variety is a projective variety with klt singularities and ample anticanonical divisor. A sequence of metric spaces converges in the Gromov–Hausdorff topology if it converges as metric spaces, while local convergence in the Cheeger–Gromov topology means smooth convergence on the regular part after choosing suitable local identifications. Let $S$ denote the singular set of a limiting variety, and let its Hausdorff codimension be the difference between the ambient dimension and the Hausdorff dimension of $S$.

Hamilton–Tian conjecture. For any global solution $\omega(t)$ as above, any sequence $(X,\omega(t))$ along the Kähler–Ricci flow contains a subsequence converging to a $\mathbb{Q}$-Fano variety $(X_{\infty},\omega_{\infty})$ in the Gromov–Hausdorff topology. The limit admits a smooth shrinking Kähler–Ricci soliton outside its singular set $S$, which is closed and has Hausdorff codimension at least $4$. Moreover, this subsequence converges locally to $(X_{\infty},\omega_{\infty})$ in the Cheeger–Gromov topology.

The conjecture describes the expected long-time geometric behavior of the normalized Kähler–Ricci flow when the initial Fano manifold does not admit a Kähler–Ricci soliton: the complex structure may degenerate, while the limit is a singular $\mathbb{Q}$-Fano variety carrying a soliton on its smooth locus. The supplied text attributes this conjecture to Tian, but gives no evidence of resolution.