TITLE: Algebraic de Rham cohomology vs. analytic de Rham cohomology
QUESTION [21 upvotes]: 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?

REPLY [8 votes]: 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.