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Alexandre Eremenko
Alexandre Eremenko
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You allow $N=k$, so $CP^2$ is a trivial example. 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.

You allow $N=k$, so $CP^2$ is a trivial example. 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 it is easy to see that the images can "fill" several planes. So $k=2$ according to your definition.

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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Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

You allow $N=k$, so $CP^2$ is a trivial example. 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 it is easy to see that the images can "fill" several planes. So $k=2$ according to your definition.