Hamilton–Tian conjecture for normalized Kähler–Ricci flows on Fano manifolds

Let $M$ be a Fano $n$-manifold and let $g(t)$ be a global solution of the normalized Kähler–Ricci flow

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

Assume that $g(0)=g_0$ has canonical Kähler class $2\pi c_1(M)$. A shrinking Kähler–Ricci soliton is the limiting geometric object described by the conjecture, allowing mild singularities.

Hamilton–Tian conjecture. The pair $(M,g(t))$ converges, at least along a subsequence, to a shrinking Kähler–Ricci soliton with mild singularities.

Here, mild singularities may mean either a singular set of codimension at least $4$, with convergence in the Gromov–Hausdorff topology, or a singular set of a normal variety, with convergence as algebraic varieties in a projective space. The conjecture is a folklore formulation also known as the Hamilton–Tian conjecture; the necessary part was proved by Tian, while the full picture and its relation to the Yau–Tian–Donaldson conjecture are part of the surrounding theory.