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Let $X$ be a smooth projective variety over $\mathbf{C}$, $\Omega^{\bullet}_X$ its algebraic de Rham cohomology.

Let $p : X_{\rm an}\to X_{\rm Zar}$ the obvious morphism of sites.

We have $p^*\Omega^{\bullet}_X = \Omega^{\bullet}_{X_{\rm an}}$, the right side being analytic de Rham cohomology of $X$.

Is the natural map

$$\Omega^{\bullet}_X\to Rp_*p^*\Omega^{\bullet}_X$$ a quasi isomorphism of Zariski sheaves on $X$?

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    $\begingroup$ You're being a bit imprecise about whether you want the whole de Rham complex or its pieces, but I think both should follow by Serre's GAGA plus coherence of higher direct images. $\endgroup$
    – Joe Berner
    Commented Apr 2, 2018 at 2:01
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    $\begingroup$ Since you are asking of a property of complexes of sheaves, the question is of purely local nature. In fact your question is equivalent to the question if algebraic de Rham cohomology coincides with analytic de Rham cohomology for smooth schemes, which is a theorem of Grothendieck if you work over a field embedded in C. $\endgroup$
    – D.-C. Cisinski
    Commented Apr 2, 2018 at 14:05

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