8 problems

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…

Non-hyperbolicity conjecture. If

Let $k$ be an algebraically closed field of characteristic zero. A variety over $k$ is strongly-pseudo-Brody hyperbolic (respectively, strongly-pseudo-Kobayashi hyperbolic) if ever…

Green--Griffiths--Lang conjecture. The following are equivalent:

Demailly's conjecture. The variety $X$ is Kobayashi hyperbolic if and only if there exists $k\in\mathbb{N}$ such that $(X,T_X)$ has negative $k$-jet curvature.

Kobayashi–Lang conjecture. The canonical line bundle $K_X$ is ample.

Let $X$ be a Kobayashi hyperbolic compact manifold of dimension $n$. Let ${\mathbb B}$ be the unit ball in $\mathbb C^p$, and let $E$ be a closed polar subset of ${\mathbb B}$. Nog…

Let $X$ be a projective variety embedded in projective space. It is approximately elliptically connected in the naïve sense if it can be connected by elliptic curves contained in a…