An entire curve on a complex variety is a holomorphic map from the complex numbers to the variety. We discuss two well-known conjectures on entire curves on very general high-degree hypersurfaces $X$ in $\mathbb{P}^n$: the Green–Griffiths–Lang Conjecture, which says that the entire curves lie in a proper subvariety of $X$, and the Kobayashi Conjecture, which says that X contains no entire curves.
We prove that (a slightly strengthened version of) the Green–Griffiths–Lang Conjecture in dimension $2n$ implies the Kobayashi Conjecture in dimension $n$. The technique has already led to improved bounds for the Kobayashi Conjecture. This is joint work with David Yang.[-]