10 problems

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.…

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…

JSS conjecture. If $X$ is a minimal model, then the cscK metric spaces near the canonical class $[K_X]$ converge geometrically to the twisted Kähler–Einstein space…

Let $n$ be a positive integer, let $v$ be a non-negative rational number, and let $cmathcal{I}subset [0,1]cap cmathbb{Q}$ be a DCC set of rational numbers. For a klt pair…

Let $(M,\omega_0)$ be a compact Kähler manifold with semi-ample canonical bundle $K_M$. Let $c_1(M)=[\operatorname{Ric}(\omega_0)]$ and consider any sequence of cscK metrics in the…

Weil–Petersson extension and completion conjecture. The metric $\omega_{WP}$ extends uniquely to a non-negative closed $(1,1)$-current on $O{S}$ with continuous local potentials, a…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle $K_X$. For sufficiently large $\ell$, let $\pi:X\rightarrow X_{can}\subset\mathbb{C}P^N$ be the…

Let $X$ be a Kähler manifold of dimension $n$ with positive Kodaira dimension strictly less than $n$, and suppose that $K_X$ is semi-ample. Let $\pi:X\rightarrow X_{can}$ be the ho…

Constancy criterion for simultaneous canonical models. The morphism $f:(X,\Delta)\to S$ has a simultaneous canonical model if and only if the function

Kähler–Ricci flow convergence conjecture. The metric spaces $(X,g(t))$ converge to the metric completion $(X_{can},g_\infty)$ in Gromov–Hausdorff topology as $t\to\infty$. This is…