6 problems

A complex manifold $Y$ is an Oka-1 manifold if it enjoys the Oka property with approximation and jet interpolation for maps $X\to Y$ from any open Riemann surface $X$. A projective…

Universal entire-curve conjecture. There should be some entire curve $g:\mathbb{C}\rightarrow X$ such that the concentric discs

A complex manifold $X$ is rationally connected if every pair of points in $X$ can be connected by a rational curve $\mathbb{CP}^1\to X$. A compact Kähler manifold is a compact comp…

A complex manifold $X$ is rationally connected if every pair of points in $X$ can be connected by a rational curve $\mathbb{CP}^1\to X$. A compact projective manifold is a compact…

Campana–Winkelmann conjecture. If $Y$ satisfies BOP, then $Y$ is special and every holomorphic map from $Y$ to a Brody hyperbolic Kähler manifold is constant.

Gromov's conjectures. The following assertions hold: