Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle

Theorem 1.1: if $X$ is a compact connected Kahler manifold of positive dimension and every holomorphic map $\mathbb C\to X$ is constant, then $K_X$ is ample and $X$ is projective. Corollaries: $K_X^{\otimes m}$ is globally generated for $m\ge n+2$ (using the release's Fujita freeness result), and a finite cover splits as a ball-quotient-type factor times a hyperbolic factor when the universal cover is semialgebraic. It does not cover singular spaces, non-Kahler compact complex manifolds, or the converse direction (Green-Griffiths-Lang type statements).

Theorem 1.1: if $X$ is a compact connected Kahler manifold of positive dimension and every holomorphic map $\mathbb C\to X$ is constant, then $K_X$ is ample and $X$ is projective. Corollaries: $K_X^{\otimes m}$ is globally generated for $m\ge n+2$ (using the release's Fujita freeness result), and a finite cover splits as a ball-quotient-type factor times a hyperbolic factor when the universal cover is semialgebraic. It does not cover singular spaces, non-Kahler compact complex manifolds, or the converse direction (Green-Griffiths-Lang type statements).