Generalized Green–Griffiths–Lang conjecture

Let $X$ be a smooth quasi-projective variety. A variety is pseudo-Picard hyperbolic if it satisfies the punctured-disk extension property outside a proper Zariski closed subset, and pseudo-Brody hyperbolic if every nonconstant entire curve has image in a proper Zariski closed subset. It is strongly of log general type if some proper Zariski closed subset contains every positive-dimensional closed subvariety that is not of log general type. Generalized Green–Griffiths–Lang conjecture. The following properties are equivalent: $X$ is of log general type; $X$ is pseudo-Picard hyperbolic; $X$ is pseudo-Brody hyperbolic; and $X$ is strongly of log general type. This conjecture connects positivity of the logarithmic canonical bundle with analytic hyperbolicity and remains open, including for surfaces.