33 problems

Let $X$ be a smooth complex projective variety. Its fundamental group is large if, for every closed irreducible positive-dimensional subvariety $Z\subset X$, the image … is infinit…

An open connected Riemann surface is a connected one-dimensional Stein manifold. Forster's conjecture. Every open connected Riemann surface embeds properly holomorphically into…

Let $\Gamma<\operatorname{Aut}(\mathbf B^n)$ be a convex-cocompact, torsion-free subgroup, and let $M_\Gamma=\mathbf B^n/\Gamma$. Suppose that $\delta(\Gamma)=2$. The non-Stein cha…

Let $X$ be a complex space with … A family of discrete subsets of $X$ is inequivalent when its members are inequivalent under the equivalence relation considered in the paper. Unco…

Let $Y$ be a contact manifold, and let $X_1$ and $X_2$ be nonsingular Stein fillings of $Y$. Let $D^+_1$ and $D^+_2$ be the Dolbeault-Dirac operators on the fillings, and define th…

Let $X$ be a Stein manifold diffeomorphic to ${\mathbb C}^n$ and having the density property, meaning that the closure of the Lie algebra generated by the complete holomorphic vect…

Let $G_\mathbb R/K$ be a Riemannian symmetric space, let $D$ be the Akhiezer–Gindikin domain, and let $C(S)_0$ denote the cycle space associated with a closed orbit $S$, while…

A special exhaustion function on a complex manifold is an exhaustion satisfying a Monge–Ampère equation of the type described in the surrounding context. Burns–Diederich conjecture…

Universality conjecture. As $\hbar\rightarrow 0$, the asymptotic behaviour of $\operatorname{dim}{\mathcal H}_{poly}$ is universal, as in the compact Kähler case.

Let $B$ be a Stein complex space and let $F$ be a Stein fiber in a holomorphic fiber bundle over $B$. Serre's conjecture. The total space of the fiber bundle is Stein. This conject…

Michael's conjecture. The existence of sequences of Fatou-Bieberbach domains having the properties needed in the reduction of Michael's conjecture should be established.

Let a Stein trace be the Stein $4$-manifold obtained from a Legendrian knot by attaching a Stein $2$-handle. Contact annulus twists modify the knot and its contact data, while Lutt…

Let $\Gamma$ be a convex-cocompact and torsion-free subgroup of $\mathrm{PU}(n,1)$ acting on the complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$, and suppose that its critic…

Let $n\geq 1$ and let $\Gamma$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$ acting on the complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$. Define the critic…

Let $W$ be a Stein space of dimension $n\geq 3$ with at most finitely many isolated normal singularities. Assume that $W$ has a compact strongly pseudoconvex smooth boundary…

Let $\mathfrak{Stein}$ and $\mathfrak{Weinstein}$ denote the spaces of Stein and Weinstein structures, respectively, and let … be the map sending a Stein structure to its associate…

Let $K$ be a knot, let $TB(K)$ denote its Thurston–Bennequin number, and let $X=X_n(K)$ be the $n$-trace on $K$ with $n<TB(K)$. Let $s(X;0)$ and $s(X;1)$ be the corresponding lasag…

Let $(X,c9)$ be a closed Kähler manifold and let $\Sigma$ be a complex hypersurface such that $X\setminus\Sigma$ is subcritical Stein. Biran–Cieliebak's uniruledness conjecture. Th…

Density-property characterization. $X$ satisfies the density property if and only if it is an $RR$-space.

Let $G$ be a simply connected unimodular solvable Lie group equipped with a left invariant complex structure. A complex manifold is Stein if it is biholomorphic to a closed complex…

Let $M$ be a compact LCK complex surface with proper potential, and let $\widetilde M$ be its Kähler $\mathbb Z$-covering. The metric-completion conjecture. The metric completion o…

Let $X$ be a complex projective manifold whose universal cover is Stein. Liu–Maxim–Wang conjecture for Stein universal covers. The cotangent bundle $T^*X$ should be nef. The paper…

Let $X$ be a possibly singular complex projective variety, let $\widetilde{X}$ be its universal cover, and let $\chi_{\mathrm{IH}}(X)$ denote its intersection-homology Euler charac…

Let $X$ be an $(n+d)$-dimensional Stein manifold, and let $M$ be a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $X$. Stein-manifold inflexibility conjecture. The result th…

Let $(X, HX, J)$ be a three dimensional strongly pseudoconvex CR manifold. A stable embedding is a CR embedding $f:(X, HX, J)\rightarrow\mathbb C^k$ such that every sufficiently sm…