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 ≥ 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.