6 problems

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 $\Omega\subset\mathbb{C}^n$ be a bounded strictly pseudoconvex domain with $C^\infty$ boundary, and let $\omega_B$ denote its Bergman metric. Cheng's conjecture. If $\omega_B$…

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…

Tangent-cone conjecture. The following properties hold:

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…

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…