Euler characteristic in Zariski vs. classical Topology Let $X$ be a smooth projective, complex variety. Denote by $\bar X$ its analytification, i.e. $X$ with the "classical" topology of a complex manifold. Now do these spaces have the same Euler characteristic $\chi(X)=\chi(\bar X)$? Why (not)?
Edit: To clarify, by $\chi(X)$ I mean the topological Euler characteristic of the underlying topological space of the scheme $X$, not the Euler characteristic $\chi(X,\mathcal{O}_X)$ of its structure sheaf.
 A: On the topological side, we have the Fröhlicher relation (where $\Omega^p_X$ is the sheaf of regular algebraic $p$-forms on $X)$
$$\chi(X^{top })=\sum_{p,q} (-1)^{p+q}dim_{\mathbb C} H^q(X,\Omega^p_X)$$
It is quite remarkable in that the left-hand side can be computed if you just know the underlying topological space of $X$, whereas the right hand side, involving coherent sheaf cohomology, requires for its computation the algebraic structure of $X$.    
On the algebraic side, given a  smooth algebraic variety $X$ of dimension $n$,  its Euler characteristic $\chi(X)$ may be taken to mean the integer $\chi(X)=\sum_{i} (-1)^{i}dim_{\mathbb C} H^i(X,\mathcal O_X)$, an invariant closely related to the arithmetic genus $p_a(X)$ by the formula 
$p_a(X)=(-1)^n(-1+\chi (\mathcal O_X) )$ .
For a curve Riemann-Roch tells you that $\chi (\mathcal O_X)=1-g$, where $g$ is the good old genus of that curve, and so we have $p_a(X)=(-1)^1(-1+(1-g))=g$, which is reassuring!
For higher dimensional varieties we have the formula ( a special case of Hirzebruch's Riemann-Roch formula)
$$\chi (\mathcal O_X)=t_{2n}[Todd(X)]$$
The right-hand side, the Todd genus, takes some time to unravel, but the amazing thing is that it only depends on the differentiable structure $X^{diff}$ of $X$, whereas the individual terms of the alternating sum on the left-hand side depend on the algebraic structure of $X$.
For $n=2$, the case of a surface, this unpacks to Max Noether's formula 
$$ \chi (\mathcal O_X)=\frac{c_1^2(X)+c_2(X)}{12}       $$
where the $c_i$'s are the Chern classes of the differentiable tangent bundle.
Note that it is a priori not in thre least evident that the right-hand side is an integer, i.e. that its numerator is divisible by $12$ !
(Emmy Noether, of noetherian fame, was Max's daughter by the way)
