The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, 245-274
In their introduction they write, that this was inspired by the following observation:
In the l-adic picture the higher massey products would go between spaces with different frobenius eigenvalues, hence had to vanish.
I have two questions about this:
Is it possible to give a proof of DGMS result (with appropriate coefficients) replacing hodge theory by l-adic sheaves?
An l-adic proof, would probably not need the assumption that the variety is projective, but only that it is proper. Is the cohomology with appropriate coefficients of any smooth proper complex variety formal?
