112 problems

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 $(\mathcal{X},\omega)$ 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 smooth Kähler manifold with nef canonical bundle, and let $g_0$ be any initial Kähler metric on $X$. Let $g(t)$ be the solution of the unnormalized Kähler–Ricci flow.…

Feldman–Ilmanen–Knopf conjecture. The set $\Sigma$ should be an analytic subvariety of $X$. This predicts that the locus where a finite-time Kähler–Ricci-flow singularity fails to…

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…

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…

Initial-metric independence conjecture. The singularity type at infinity is independent of the initial metric $\omega_0$.

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

Let $M$ be a $\mathbb Q$-Fano group compactification, and consider the Kähler-Ricci flow on $M$ with initial metric $\omega_0$. Suppose its limit $M_\infty$ is obtained through two…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle, let $\pi:X\rightarrow X_{can}$ be the associated holomorphic Calabi–Yau fibration, and let $S$…

Let $X$ be a Fano projective variety and consider the unnormalized Kähler-Ricci flow, allowing analytic surgery as singularities develop. Tian's extinction conjecture. The flow bec…

Let $M$ be a higher-dimensional Kähler manifold, and consider a continuous path of Kähler metrics or related geometric data on $M$ along which the strong partial $C^0$-estimate is…

Let $M^n$ be a compact $n$-dimensional Kähler manifold with positive first Chern class $C_1(M)$, and consider the normalized Ricci flow … A gradient Kähler–Ricci soliton is a solut…

Let $X$ be a nonsingular variety of dimension $n$ with canonical model $X_{\mathrm{can}}$ and pluricanonical fibration $f:X\rightarrow X_{\mathrm{can}}$. In the intermediate case…

Compactness conjecture. There are no noncompact Type I limit solutions with nonnegative bisectional curvature and positive Ricci curvature.

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…

Sun–Zhang's conjecture. Every Kähler–Ricci shrinker has a unique tangent space at infinity, and this limit is a metric cone.

Interval-wise completeness conjecture. For $t\in(t_{1+\ell},t_{\ell})$, the metric $\omega_t$ is complete on $\Omega_{\ell}$ and is a Kähler current on $X$. This refines the propos…

Let $(X,\omega)$ be a compact Kähler manifold, let $\varphi_0$ be an $\omega$-plurisubharmonic function, and let $\omega_t=\omega+dd^c\varphi_t$ be the solution of the twisted Kähl…

Let $(X,\omega)$ be a compact Kähler manifold, let $\varphi_0$ be an $\omega$-plurisubharmonic function, and let $\omega_t=\omega+dd^c\varphi_t$ solve the twisted Kähler-Ricci flow…

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 $X$ be a compact Kähler manifold undergoing the normalized Kähler–Ricci flow, and suppose that, in the finite-time singularity case, there is a proper holomorphic map with conn…

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…