Holomorphic injectivity and the Hopf map

Details Holomorphic injectivity and the Hopf map Holomorphic injectivity and the Hopf map

2005-01-13

arxiv.org/abs/math/0501196v1

Geom. Funct. Anal. 14 (2004), no. 6, 1339--1351.

Mathematics

Algebraic Geometry

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

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Holomorphic injectivity and the Hopf map does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Holomorphic injectivity and the Hopf map, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Holomorphic injectivity and the Hopf map will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-274438