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What's the diffencedifference between a real manifold and a smooth variety?

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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 $U$, then there exists a global function $\tilde{g}$ that agrees with $f$ on $V$ and is identically zero outside of $U$.

By contrast, if consider the same set-up when $X$ is a non-singular variety and $f$ is a regular functionon . It may be impossible find a (non-empty) open subset global regular function $U \subset X$g$that agrees with$f$on$V$. When$g$exists, then there it is at most one extension$\tilde{f}$of unique and (when$f$to a regular function is non-zero) is not identically zero on outside of$X$. There are examples where no extension exists.U$.

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.

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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. If Suppose that $X$ is a real manifold and $f$ is a smooth function on an open subset $U U$. Given $V \subset X$, U$compactly contained in$U$, then there is exists a global function$\tilde{f}$on$X$\tilde{g}$ that restricts to agrees with $f$ on $U$. In fact, there many extensions. For example, if $V$ and is a second open subset, $U \subset V$, the function $\tilde{f}$ may be choen to be identically zero outside of $V$. U$. By contrast, if$X$is a non-singular variety and$f$is a regular function on a (non-empty) open subset$U \subset X$, then there is at most one extension$\tilde{f}$of$f$to a regular function on$X$. There are examples where no extension exists. 3. If$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.