9 problems

Let $X$ be a smooth Fano manifold admitting a Kähler–Ricci soliton, and let $Y:=K_X^{\times}$ be the complement of the zero section in its canonical bundle. A Calabi–Yau cone struc…

Existence conjecture. The conclusions of Theorem 1 hold for $M$, depending on the sign of $c_1(\det E\otimes K_B^{\vee})$. If $d_\ell=0$ and $\det E\otimes K_B^{\vee}\to B$ is triv…

Let $(M,J,g,X)$ be a simply connected complete steady gradient Kähler–Ricci soliton, meaning that $g$ is a complete Kähler metric and $X$ is a vector field generating the soliton.…

Properness conjecture. The moment map $\bm\mu$ is proper.

Let $(M^4,J)$ be a compact Fano surface admitting a Kähler–Ricci soliton $(\omega_{KRS},X)$. Let $\omega_0\in c_1$ be a Kähler metric invariant under the one-parameter subgroup…

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

Modified F-functional geodesic stability conjecture. The following are equivalent:

Sasaki–Einstein circle-bundle conjecture. If there exists a Kähler–Ricci soliton on $M$, then the $S^1$-bundle associated to $K_M$ admits a Sasaki–Einstein metric with a suitable c…

Generalized Hamilton-Tian conjecture. As $s\to\infty$, the flow starting from $\omega$ converges in the Gromov-Hausdorff sense to a Kähler-Ricci soliton $(X_\infty,\omega_{KR})$. T…