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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?

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up vote 2 down vote accepted

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.

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Take product of a hyperbolic manifold and a Riemann surface.

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@Misha So, $\mathbb C \times (\mathbb C-\{0,1\})$ is 2-hyperbolic but not Brody hyperbolic? – Saurabh T Nov 24 '12 at 20:34
Yes, of course. – Misha Nov 25 '12 at 5:14

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