15 problems

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 $X$ 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…

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 $n\geq 1$ 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…

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

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…

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 $Y$ be a projective manifold, and let $\widetilde{Y}$ be its universal cover. Nef cotangent conjecture. If $\widetilde{Y}$ is a Stein manifold, then the cotangent bundle…

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 $(X, HX, J)$ 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…

Let $X$ be a Stein manifold of dimension $n\ge 2$ equipped with a Riemannian metric. A nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces is a foliat…

Let $X_{n,k}$ be the compact contractible oriented smooth $4$-manifold whose boundary is Stein fillable. Nonexistence conjecture. The manifold $X_{n,k}$ does not admit any Stein st…