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:

  1. Is it possible to give a proof of DGMS result (with appropriate coefficients) replacing hodge theory by l-adic sheaves?

  2. 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?

  • 2
    There is a $p$-adic proof of formality for smooth proper complex varieties in the paper "F-isocrystals and homotopy types" by Martin Olsson. – ulrich Apr 30 '12 at 15:34
  • 4
    Also, see Deligne, Conjecture de Weil II, Cor 5.3.7. So the answer, I believe, is yes. – Donu Arapura Apr 30 '12 at 15:38
  • Thank you very much for your comments, I will check out these sources! – Jan Weidner Apr 30 '12 at 16:09

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