Algebraic de Rham cohomology vs. analytic de Rham cohomology Let $X$ be a nice variety over $\mathbb{C}$, where nice probably means smooth and proper.
I want to know: How can we show that the hypercohomology of the algebraic de Rham complex agrees with the hypercohomology of the analytic de Rham complex (equivalently the cohomology of the constant sheaf $\mathbb{C}$ in the analytic topology)? Does this follow immediately from GAGA? If not, how do you prove it?
I think that this does not follow immediately from GAGA because, while the sheaves $\Omega_X^i$ are coherent, the de Rham $d$ is not a map of coherent sheaves (it is not multiplicative). Am I correct in my thinking?
 A: Various people have answered the question, and also brought up some of the subtleties in applying GAGA. So I won't  rehash all that. So let me just suggest the additional reference: 
Deligne, Équations différentielles à points singuliers réguliers
especially Chapter II, section 6. These issues are dealt with carefully in a more
general setting of de Rham cohomology with coefficients in a regular integrable connection.
The result is no doubt true for a regular holonomic D-module, and it would be nice if
someone wrote this down carefully. But perhaps I'm straying too far from the original topic.
A: This does follow from GAGA via the spectral sequences associated to the dumb filtrations on the algebraic and analytic de Rham complexes of sheaves, see p. 96 of tome 29 of PMIHES in a paper of Grothendieck (1966).
A: If $X$ is smooth and proper, GAGA does in fact suffice (despite the observation that $d$ is not $\mathcal{O}_X$-linear:  One obtains a comparison map of hypercohomology spectral sequences; it is an isomorphism on the $E_2$ page by GAGA, and thus on the $E_\infty$ page.
It is to prove the general case (i.e., $X$ smooth but not necessarily proper) that one needs to do additional work.
A: I don't think you can get this directly from GAGA. The reference that I know for this result is Grothendieck, On the de Rham cohomology of algebraic varieties. It is short, beautiful, and in English.
