Question: Let $X$ be a $\mathbb{C}$-variety and let $X\_{cx}$ denote the topological space formed by its $\mathbb{C}$-points with the complex topology (i.e. the associated analytic space).

If $X$ is projective, then $X\_{cx}$ is a compact topological space. I was thinking about the converse and found a GAGA-cite that $X\_{cx}$ is compact iff $X$ is proper. This was not what I expected: If you draw for example the *real points* of the variety defined by $X^2+Y^2+Z^2=1$, you get a $2$-sphere which is nice and compact.

I can think about $\mathbb{C}$-points honestly only for curves since $\mathbb{R}^{\geq 4}$ is beyond of what I can imagine as a whole (and compactness is not a local property). Therefore, if the GAGA-cited theorem is true, there must be a big difference in relation to compactness between the real and the complex picture.

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> Why are there no non-trivial affine $X$ with $X\_{cx}$ compact (a proper affine morphism is finite) and where is (morally) the difference to the real picture?
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Answer: 
The complex points of a $\mathbb C$-scheme suffice to reconstruct the scheme.  

More precisely there is an equivalence of categories between these schemes (more precisely integral schemes of finite type over $\mathbb C$) and classical varieties, obtained by sending the scheme $X$ to its set $X\_{cx}$ of $\mathbb C$-points , endowed with the restriction of the structural sheaf: $\mathcal O\_{X\_{cx}}=O\_X|X\_{cx}$. 

The root of this equivalence resides in two facts:  

$\bullet$ The ring $\mathbb C[T\_1,\ldots , T\_n]$ is Jacobson, meaning that every prime ideal is the intersection of the maximal ideals containing it.  

This is true for any polynomial ring but the following second fact necessitates algebraic closedness of $\mathbb C$: 

$\bullet \bullet$Every maximal ideal $\mathfrak m \subset \mathbb C[T\_1,\ldots , T\_n]$ is given by a point $a=(a\_1,\ldots, a\_n)$: $\mathfrak m=(T\_1-a\_1,\ldots , T\_n-a\_n)$ 

This is completely false for $ \mathbb R[T\_1,\ldots , T\_n]$ and is the moral reason why the $\mathbb R$- points of a $\mathbb C$-scheme are vastly insufficient for determining the scheme: just think of the real points of $x\_1^2+\ldots+x\_n^2+x\_{n+1}^2=0$ in $\mathbb P^n\_\mathbb R$ [Actually there is literally nothing to think about :-)]
The following argument gives a rough idea for the non-compactness of affine $\mathbb{C}$-curves: let $X$ be such a curve and let $Y$ be its projective closure. Then $Y\setminus X$ is a finite set; take some point $x$ in this set. Chose a family $(A\_i)\_{i\in\mathbb{N}}$ of closed (in the fine topology) subsets $A\_i\subseteq Y$ with the following properties: $A\_i\supseteq A\_{i+1}$, $A\_i\neq A\_{i+1}$, $\bigcap\_i A\_i =\{x\}$.

Then $(X\setminus A\_i)\_i$ is an open covering of $X$ that possesses no finite subcovering.