10 problems

Let $X$ be a Fano manifold with a Kähler–Einstein metric $\omega_0$. Let $\mathcal{H}(X,\omega_0)/\mathbb{R}$ be the space of Kähler potentials modulo constants, and let…

Let $(X,\Delta)$ be a log Fano manifold admitting a Kähler–Einstein metric. Let $Z_{\boldsymbol{m}_{N}}$ be the K-reduced partition function, let $\mathcal{M}$ be the Mabuchi funct…

Let $\Omega$ be a bounded domain that is not a product domain. A domain is homogeneous if its automorphism group acts transitively, meaning that for any $x,y e Omega$ there is a…

Let $X$ be a Fano variety and, for each sufficiently large positive integer $k$, let … Let $\mathcal{D}_k$ be the anti-canonical effective $\mathbb{Q}$-divisor on $X^{N_k}$ whose s…

Tangent-cone conjecture. The following properties hold:

Let $(X,D)$ be a two-dimensional klt pair with $D=\sum_{j=1}^{n}(1-\beta_j)D_j$, and let $g$ be a weak conical Kähler–Einstein metric on $(X,D)$. Let $S=\operatorname{Sing}(X,D)$ b…

Let $M_\infty$ be a Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, let $o\in M_\infty$, and let $C=C_oM_\infty$ be a metric tangent cone at $o$. Let…

Ricci-threshold equality conjecture. The twisted Yau–Tian–Donaldson conjecture in this case states that

Let $X$ be a del Pezzo surface with at most canonical singularities. Del Pezzo Kähler–Einstein classification conjecture. The orbifold $X$ is Kähler–Einstein if and only if one of…

Let $X$ be a Fano variety with at most quotient singularities, and for each $n\in\mathbb{N}$ let $\mathrm{lct}_{n,2}(X)$ be the infimum of the log canonical thresholds of pencils i…