10 problems

Let $M$ be a compact Kähler manifold with negative sectional curvature, and let $\widetilde{M}$ denote its universal covering. Siu's ball uniformization conjecture. The universal c…

Let $\Omega \subset \operatorname{\mathbb{C}}^d$ with $d \geq 2$ be a bounded pseudoconvex domain whose boundary is $C^2$. Suppose that $\Omega$ admits an open quotient of finite v…

Let $D$ be an irreducible bounded domain with a one-parameter group of holomorphic automorphisms and a compact quotient. Kazhdan's conjecture. The domain $D$ is biholomorphic to a…

Let $\Omega\subset \mathbb C^n$ be a bounded homogeneous domain, let $m\in\mathbb N^*$, and let $\Omega_{m,s}$ be the associated Hartogs domain, with $s\neq 0$. Bergman–Kähler–Ricc…

Let $\mathbb{H}$ denote the hexablock, let $G(\mathbb{H})$ be the group of explicitly given automorphisms generated by the maps described above, and let…

Let $M$ be a hyperbolic complex surface, and let $\Omega\subset\subset M$ be a subdomain with $C^{1,1}$ boundary. A compact quotient of $\Omega$ means that $\Omega$ admits a quotie…

Holomorphic equivalence conjecture. If there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$, then $D_1$ and $D_2$ are holomorphically equivalent.

Let $D$ be a bounded convex domain with a finite-volume manifold quotient. Finite-volume quotient conjecture. The domain $D$ is biholomorphic to a bounded symmetric domain. This is…

Bounded-domain conjecture. The manifold $M$ is biholomorphic to a bounded domain in $\mathbb{C}^n$.

Let $D\subset \mathbb{C}^n$ be a bounded pseudoconvex domain with reasonably smooth boundary, and consider its complete Cheng–Yau Kähler–Einstein metric and canonical Bergman metri…