65 problems

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 $\mathbb{Q}$…

Metric cone conjecture. If the Kähler–Ricci shrinker has maximal volume growth, then $(Z_0,d^Z_0)$ is a metric cone and coincides with the metric completion of…

Polarized affine cone conjecture. The space $Z_0$, with the topology induced by $d^Z_0$, admits a polarized affine cone structure, and the map $\Phi$ realizes the polarized Fano fi…

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold, let $\alpha$ be a closed nonnegative $(1,1)$-form, and let $(\omega(t),\alpha(t))$ solve the coupled flow on its maximal int…

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold and let $\alpha$ be a closed nonnegative $(1,1)$-form satisfying … Let $(\omega(t),\alpha(t))$ solve the Kähler–Ricci flow co…

Let $X$ be a K-semistable Fano manifold, and let $\omega$ denote the initial Kähler form for the Kähler–Ricci flow. The iterated balancing filtration is the filtration referred to…

Let $(X,d_X)$ be the singular Kähler–Ricci shrinker arising as the metric soliton modeled on the $\mathbb{F}$-limit described in the paper, with regular part…

Unboundedness conjecture. Under the assumptions above, the quantity

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 … Let $f:X\to Y$ be the canonical-mo…

Let $(M^n,g)$ be a complete noncompact Kähler manifold with nonnegative curvature and maximal volume growth. Let $g(t)$, $t\in[0,+\infty)$, be the unique complete solution of the K…

Type I singularity conjecture. Every finite-time singularity of the Kähler-Ricci flow on a Kähler surface is type I.

Tangent flows at infinity conjecture. Given a non-collapsed ancient Kähler-Ricci flow $(X,\omega(t))$, $t\in(-\infty,0]$: 1. $X$ admits a non-collapsing weak Fano fibration structu…

Finite-time two-step degeneration conjecture. For any $p\in Y$ and $q\in\pi^{-1}(p)$: 1. The Kähler-Ricci flow $\omega(t)$ naturally induces a weighted-volume-minimizing, hence K-s…

Vanishing-Lelong-number and boundedness conjecture.

Canonical limit current conjecture. There is a quasi-plurisubharmonic function $\varphi_\infty$ such that

Song–Tian's Gromov–Hausdorff conjecture.

Let $X$ be a compact Kähler manifold with semiample canonical bundle $K_X$ and intermediate Kodaira dimension $0<m<\dim X$, let $\omega_0$ be a Kähler metric on $X$, and let…

Let $g(t)$ be the maximal Kähler-Ricci flow on $X\times[0,1)$, let $\theta_Y\in\vartheta$ be a smooth closed $(1,1)$-form on $Y$, and let $p_t$ be a Ricci vertex associated to…

Let $g(t)$ be the maximal Kähler-Ricci flow on $X\times[0,1)$, let $\Phi:X\to Y$ be the limiting fibration, and let $\mathcal{F}_q=\Phi^{-1}(q)$. A Ricci vertex associated to a smo…

Let $\mathcal{X}$ be a metric-flow limit of Type I blow-ups of a Kähler-Ricci flow, with time slices $(\mathcal{X}_t,d_t)$ as in Theorem 2main1. A quasi-projective normal variety i…

Let $g(t)$ be the smooth solution of the Kähler-Ricci flow on $X\times[0,T)$, where $T<\infty$, and let $\Phi:X\to Y$ be the map induced by the limiting semi-ample class. For…

Tosatti's conjecture. The singularity type at infinity does not depend on the choice of the initial metric $\omega_0$.

Let $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ be the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at a fixed point $p$ of the standard torus action, and let…

Let $\omega=\omega(t)$ be a solution of the Kähler-Ricci flow on a compact manifold $X$, defined on its maximal time interval $[0,T)$. Here ${\rm diam}(X,\omega(t))$ denotes the di…

Let $M^m$ be a compact Kähler manifold with semiample canonical bundle and Kodaira dimension $n=\kappa(M)$ satisfying $0<n<m$. Let $f:M\to N$ be its Iitaka fibration onto a normal…