Is it not sufficient to prove the result for the projective space?

Thank you for the reference, it's very interesting. The author leaves open the question whether there exists an algebraically simply connected algebraic variety which is not topologically simply connected. Such an example would be much more surprising!

So the statement is that for Riemann surfaces, the topological group injects into the algebraic fundamental group (the profinite completion of the former). If it is OK to ask questions in comments, I would like to know if this holds more generally for smooth complex algebraic varieties.

I don't know the answers to your questions, but there is a section on this in Brion, Kumar "Frobenius Splitting Methods in Geometry and Representation Theory" (5.2, p.169) where they reprove (some of) Broer's results in all characteristics.

It is indeed strange that in the Brion-Kumar book, the main reference on the subject, they prove Kodaira vanishing but don't even remark that the full Kodaira-Akizuki-Nakano vanishing holds.

Great! Why don't you post this as an answer?

Hmm, what if the principal bundle is not locally trivial in our topology? I heard that there are principal bundles which are not étale-locally trivial.

Yes, they are, by the Bruhat decomposition.

If you don't want to use intersection theory, note that the function $y\mapsto \chi(X_y, \mathcal{M}_y)$ is locally constant (Hartshorne III 12). Then use Riemann-Roch.

It fails even in char.0, because the Poincare lemma works only formally locally and the de Rham complex is not exact in positive degrees. On the other hand, the cohomology of any constant sheaf vanishes on an irreducible topological space, so anyway taking resolutions of $k$ would not lead no anything of interest.

I guess I didn't understand the argument of Deligne-Illusie, that we can reduce to positive characteristic in such a way that the special fiber lifts to $W_2$. I thought that if we have a lift to char. 0 then we have a lift to $W_2$ by some universal property of the Witt vectors :). Now I read it more carefully...

This is subtle, because the cases are new only for $p\leq\dim X$, otherwise we can use Deligne-Illusie...