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I stumbled upon the following remark somewhere and unfortunately lacks a proof. "Every Riemann Surface in $\mathbb{C}^{2}$ in non-compact". A complex algebraic curve given as a zero locus of polynomial $P \in \mathbb{C}[x,y]$, apparently is not compact (since it's not bounded). But what about other Riemann Surfaces in $\mathbb{C}^{2}$, can you give me a reference or a proof of the aforementioned statement? Moreover, does this sentence mean that every Riemannian Surface cannot admit a holomorphic embedding into $\mathbb{C}^{2}$ even if it is compact on its own as a topological space?

P.S. If I have said something completely wrong or mistaken please do let me know. I'm not very familiar with all these things. Thank you!

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  • $\begingroup$ A smooth embedding would not be a problem, as $\mathbb{C}^2$ is diffeomorphic to $\mathbb{R}^4$. I think the problem here is that you want a holomorphic embedding. $\endgroup$ – Daniel Robert-Nicoud Nov 28 '16 at 12:37
  • $\begingroup$ you're right! I just edited it! $\endgroup$ – user321268 Nov 28 '16 at 12:39
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    $\begingroup$ There are no non-constant maps $X\to\mathbb{C}^2$ if $X$ is compact. Indeed, one would then obtain a non-constant coordinate function $X\to\mathbb{C}$ but every holomorphic function on $X$ is constant (recall that the image of a non-constant map is both open (open mapping theorem), connected, and closed (since it's compact!) which implies it's $\mathbb{C}$ which is not compact). $\endgroup$ – Alex Youcis Nov 28 '16 at 12:46
  • $\begingroup$ Thank you Alexis! You're right, it was almost trivial! Perhaps you can write it as answer I think! $\endgroup$ – user321268 Nov 28 '16 at 13:00
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    $\begingroup$ Possible duplicate of Embedding of Kähler manifolds into $\Bbb C^n$ $\endgroup$ – Andrew D. Hwang Dec 19 '16 at 13:05

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