Hyperbolicity implies general type conjecture

Let $X$ be a compact complex manifold of dimension $n$. It is hyperbolic when it has no non-constant entire holomorphic curves, or in the stronger Kobayashi sense described in the source. Its canonical bundle is

$$K_X=\bigwedge^nT^*X.$$

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 dimension.

This conjecture expresses the expected link between complex hyperbolicity and positivity of the canonical bundle. The source presents it as an open conjecture and gives no resolution.