Holomorphic injectivity and the Hopf map

Abstract

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 Möbius 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.