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 $n\ge 2$, 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.

LaTeX, 10 pages, to appear in Geom. and Funct. Anal

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