\section{Introduction}

Let $X$ be a compact complex manifold. An \emph{entire curve} traced in $X$ is by definition a non constant holomorphic map $f\colon\mathbb C\to X$. By Brody's criterion $X$ is Kobayashi-hyperbolic if and only if $X$ does not admit any entire curve.

At the very beginning of the theory, in the early 70's, very few examples of (higher dimensional) compact complex manifolds where known: mainly compact quotients of bounded domains in $\mathbb C^n$. Suppose to be in the smooth case, and consider a compact complex manifold $X$ whose universal cover is a bounded domain $\Omega$ in $\mathbb C^n$. Then, since $\Omega$ admits the Bergman metric $\omega_B$, and this metric is invariant BLABLABLA [Kobayashi]. In particular, $K_X$ is ample and $X$ is projective.