Green--Griffiths--Lang conjecture for projective varieties

Let $X$ be a projective variety over

. A projective variety is **groupless** if every morphism from a connected algebraic group to it is constant; $X^{\operatorname{an}}$ denotes its associated complex analytic space. A variety is of general type when its canonical birational geometry has maximal Kodaira dimension. **Green--Griffiths--Lang conjecture.** The following are equivalent: 1. $X$ is groupless over

. 2. $X^{\operatorname{an}}$ is Kobayashi hyperbolic. 3. Every closed subvariety of $X$ is of general type.

This conjecture is the broader framework from which the paper derives Demailly's conjecture. It remains open in the stated generality.