arXiv math/05011962005-01-13math.AGmath.DG

Holomorphic injectivity and the Hopf map

Scott Nollet, Frederico Xavier

备注：: LaTeX, 10 pages, to appear in Geom. and Funct. Anal
发表信息：: Geom. Funct. Anal. 14 (2004), no. 6, 1339--1351
DOI：: 10.1007/s00039-004-0494-3

英文摘要：

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.

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