Welcome to MathOverflow

Question

1

A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. Can you give an example of complex $2$-hyperbolic manifold?

edited Nov 24 at 9:20

asked Nov 24 at 8:39

Saurabh T
37●4

Alexandre Eremenko · Accepted Answer · 2012-11-24 14:56:36Z UTC

A non-trivial example is $CP^3$ minus 5 hyperplanes in general position. According to a theorem of Borel every holomorphic image of $C$ in this manifold is contained in a plane. (And there are finitely many of these planes). So every image of $C^2$ is also contained in a plane. And it is easy to see that the images can "fill" several planes. So $k=2$ according to your definition.

Misha · Answer 2 · 2012-11-24 14:43:48Z UTC

0

Take product of a hyperbolic manifold and a Riemann surface.

answered Nov 24 at 14:43

Misha
12.6k●1●26●48

	@Misha So, $\mathbb C \times (\mathbb C-\{0,1\})$ is 2-hyperbolic but not Brody hyperbolic? – Saurabh T Nov 24 at 20:34
	Yes, of course. – Misha Nov 25 at 5:14

k-Hyperbolic manifolds

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.