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This feels like a stupid question, but I am missing some imagination here.

Anyway, I know that the Riemann sphere $\mathbb{C}\cup\{\infty\}$ is a one-dimensional complex manifold and that I can also view it as a 2-sphere embedded as a real submanifold into $\mathbb{R}^3$.

However, is the Riemann sphere also an (immersed or embedded?) one-dimensional complex submanifold of $\mathbb{C}^2$?

Somehow, since $\mathbb{C}^2$ has one (real) dimension more than $\mathbb{R}^3$, I feel it should work. Does it?

What about a (one-dimensional complex) Torus?

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There is no compact complex analytic submanifolds $M$ in $\mathbb C^n$ (except for points), since the coordinate functions $(z_1, \cdots z_n)$ restrict to complex analytic functions on it, which has to be constant as $M$ is compact.

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    $\begingroup$ Well, there are complex analytic submanifolds in $\mathbb{C}^n$, but they are all of dimension zero. $\endgroup$ – Najib Idrissi Sep 14 '16 at 9:00
  • $\begingroup$ @NajibIdrissi Edited. ${}{}$ $\endgroup$ – user99914 Sep 14 '16 at 9:01
  • $\begingroup$ Sorry, I don't yet understand. I have the coordinate functions $z_i$ each mapping $M$ into $\mathbb{C}$ analytically. Why are those constant? Can you apply Liouvilles Theorem somehow? $\endgroup$ – TheAbelian Sep 14 '16 at 9:30
  • $\begingroup$ Something like that @TheAbelian, I am thinking of the maximum modulus principle. $\endgroup$ – user99914 Sep 14 '16 at 9:32
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    $\begingroup$ By coordinate function I mean the one on $\mathbb C^n$ (that is, $(z_1, \cdots z_n) \mapsto z_i$). Just restrict these functions to $M$, the compactness of $M$ implies that $|z_i|$ attains a maximum, but this is impossible since they are holomorphic functions. @TheAbelian $\endgroup$ – user99914 Sep 15 '16 at 3:54

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