I need the proof (reference) of Mori’s theorem about this implication : Let $X$ be a projective complex manifold. If $X$ contains no rational curves, then $K_K$ is nef.
A more precise form of this result is originally due to Miyaoka and Mori, see Theorem 3.6 page 67 in
Olivier Debarre: Higher-dimensional algebraic geometry, Universitext. New York, NY: Springer. xiii, 233 p. (2001). ZBL0978.14001.