14 problems

Let $\Omega$ be a normal Stein space with a compact, smooth, strongly pseudoconvex boundary. Let $\Omega{\operatorname{Reg}}$ denote its regular part, and let the Bergman metric be…

A domain in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a connected open set. A circle domain is a domain whose complementary components are points or close…

Sasakian uniformization conjecture. Then $M$ is CR biholomorphic to the standard Heisenberg group $\mathbb{H}_{n}$.

Yau's uniformization conjecture. If $M$ has these properties, then $M$ is biholomorphic to the standard $m$-dimensional complex Euclidean space

Let $(M,\omega)$ be a compact $n$-dimensional Kähler manifold with positive holomorphic bisectional curvature. Frankel's conjecture. Then $M$ is biholomorphic to $\mathbb{CP}^{n}$.…

Countable invariant uniformization conjecture. Let $E$ be a Borel equivalence relation on a Polish space $X$. Then the following are equivalent: (a) $E$ is reducible to countable;…

Let $(M,g)$ be a compact Hermitian manifold with Griffiths non-negative Chern curvature and positive first Chern--Ricci curvature. A rational homogeneous manifold is a quotient…

Uniformizability conjecture. The stack $\mathcal{M}_h$ is uniformizable by a quasi-projective scheme over $\mathbb{Q}$. Uniformizability would allow finiteness results proved for s…

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold with positive pseudohermitian bisectional curvature. Let $\mathbf{H}_n=\mathbb{C}^n\times\mathbb{R}$ be the standard…

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold with nonnegative pseudohermitian bisectional curvature, and let $\mathcal{O}_p^{CR}(M)$ denote the ring of all CR-holomo…

Let $X$ be a compact Kähler manifold of dimension $n$ with generically large fundamental group, meaning that the fundamental-group image of every subvariety through a very general…

Iitaka's conjecture. Then $X$ is a torus, up to finite étale cover.

Assume the setup of a parahoric group scheme $\mathcal G$ over a curve $X$ and let $\mathcal M_{\mathcal G/X}$ denote the moduli stack of $\mathcal G$-torsors. Let $x\in X(k)$, let…

Let $X$ be a Riemann surface of genus $g$, let $D$ be an effective divisor of degree $d$, and let…