TITLE: What's the difference between a real manifold and a smooth variety?
QUESTION [42 upvotes]: I am teaching a course on Riemann Surfaces next term, and would like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties (over, say, $\mathbb{C}$).  I am looking for examples that would be meaningful to 2nd year US graduate students who has taken 1 year of topology and 1 semester of complex analysis.
Here are some examples that I thought of:
1.  Every $n$-dimensional real manifold embeds in $\mathbb{R}^{2n}$.  By contrast, a projective variety does not embed in $\mathbb{A}^n$ for any $n$.  Every $n$-dimensional non-singular, projective variety embeds in $\mathbb{P}^{2n+1}$, but there are non-singular, proper varieties that do not embed in any projective space.
2.  Suppose that $X$ is a real manifold and $f$ is a smooth function on an open subset $U$.  Given $V \subset U$ compactly contained in $U$, there exists a global function $\tilde{g}$ that
agrees with $f$ on $V$ and is identically zero outside of $U$.
By contrast, consider the same set-up when $X$ is a non-singular variety and $f$ is a regular function.  It may be impossible find a global regular function $g$ that agrees with $f$ on $V$.  When $g$ exists, it is unique and (when $f$ is non-zero) is not identically zero on outside of $U$.
3.  If $X$ is a real manifold and $p \in X$ is a point, then the ring of germs at $p$ is non-noetherian.  The local ring of a variety at a point is always noetherian.  
What are some more examples?
Answers illustrating the difference between real manifolds and complex manifolds are also welcome.

REPLY [6 votes]: Some embedding statements.
A compact complex subvariety of ${\mathbb{C}}^n$ is a point. However, every compact real manifold of dimension $n$ can be realized as a submanifold of some ${\mathbb{R}}^{2n}$.
There are compact complex manifolds that cannot be embedded into complex projective space. An example most often quoted in textbooks is the Hopf manifold, which is not even Kahler. On the other hand, I heard that embedding into real projective space is not often considered in differential geometry.