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?


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.

| cite | improve this answer | |
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.