8 problems

Let $(M,[\omega])$ be a compact Kähler manifold with a fixed Kähler class, and let $Ca$ denote the Calabi energy. Suppose the class admits an extremal Kähler metric, whose Calabi e…

Let $M$ be a compact Kähler manifold and fix a Kähler class. Let $Ca$ denote the Calabi energy and ${\mathbf E}$ the K-energy functional. Lower-bound conjecture for the K-energy. I…

Let $(X,L)$ be a polarized variety. Every $d_2$-bounded weakly convergent sequence in $\mathcal{E}_{\mathrm{NA}}^2(X,L)$ should be $d_1$-convergent. This conjecture would provide t…

Assume $(X,L)$ is klt. For $1\le p<2$, equip $\mathcal{E}_{\mathrm{NA}}^2(X,L)$ with the $d_p$-topology. Then the normalized superlevel set … should be compact in the $d_p$-topolog…

Let $(X,L)$ be a polarized variety and let $\varphi\in\mathcal{E}_{\mathrm{NA}}^2(X,L)$ subordinate to its maximal geodesic ray $\{\ell_{\varphi,t}\}_{t\in[0,\infty)}$. The entropy…

Suppose $(X,L)$ has only klt singularities. The non-archimedean Calabi energy should satisfy … with an appropriate definition of the right-hand side. This would identify the non-ar…

Let $(X,L)$ be a polarized normal variety with only klt singularities. The normalized superlevel set … should be compact in the $E_{\exp}$-topology. This compactness is proposed as…

Let $(M,J)$ be a compact complex surface, and let $g$ be a Bach-flat Kähler metric, meaning a Kähler metric compatible with $J$ whose Bach tensor vanishes. Let $\mathcal{C}$ denote…