Une g\'en\'eralisation du th\'eor\`eme de Kobayashi-Ochiai

Let $\phi:\Bbb C^n\to X$ a holomorphic map to an $n$-dimensional connected compact complex manifold $X$. We establish links between the positivity properties of the canonical bundle of $X$ and the rate of growth of $\phi$ which extend results of Kodaira and Kobayashi-Ochiai. For example: if the average degree of $\phi$ on balls of radius $r$ grows slowlier than $r^{2n}$, then $K_X$ is not pseudo-effective. If $X$ is moreover projective, it is uniruled. Assuming now that $K_X$ is pseudoeffective, of numerical dimension $\nu$, we show that the characteristic function of $\phi$ grows at least as fast as $r^{((2n)/(n-\nu))}$.