7 problems

Hyperbolicity conjecture for general hypersurfaces. A general hypersurface

Green-Griffiths-Lang conjecture. Every entire curve $f: \mathbb{C} \to X\setminus D$ is algebraically degenerate; that is, the image of $f$ is contained in a proper algebraic subva…

Picard–analytic equivalence conjecture. The manifold $X$ is $b[31m\ellb[0m$-Picard hyperbolic if and only if it is $b[31m\ellb[0m$-analytically hyperbolic.

Extension property. Every non-degenerate holomorphic map

Lang's conjectures. If $X$ is hyperbolic, then it is mordellic. Moreover, $X\setminus\operatorname{Exc}(X)$ is mordellic.

Hyperbolicity implies general type conjecture. If $X$ is hyperbolic, then $X$ should be of general type; equivalently, $K_X$ should be big and $X$ should have maximal Kodaira dimen…

Campana's core-distance conjecture. The Kobayashi pseudo-distance is the pullback of the orbifold Kobayashi pseudo-distance on the core: