  We give sharp conditions on a local biholomorphism $F:X \to \mathbb C^{n}$
which ensure global injectivity. For $n \geq 2$, such a map is injective if for
each complex line $l \subset \mathbb C^{n}$, the pre-image $F^{-1}(l)$ embeds
holomorphically as a connected domain into $\mathbb C \mathbb P^{1}$, the
embedding being unique up to M\"obius transformation. In particular, $F$ is
injective if the pre-image of every complex line is connected and conformal to
$\mathbb C$. The proof uses the topological fact that the natural map $\mathbb
R \mathbb P^{2n-1} \to \mathbb C \mathbb P^{n-1}$ associated to the Hopf map
admits no continuous sections and the classical Bieberbach-Gronwall estimates
from complex analysis.
