Grothendieck's generalized Hodge conjecture has the following special case: if a smooth proper variety X over C has no global i-forms, for some i, then there exists a non-empty Zariski open U such that the restriction map on i-th singular cohomology H^i(X)->H^i(U) is zero.

I will describe an approach to checking this consequence via p-adic Hodge theory. We prove that for a smooth proper scheme X over Z_p with H^0(X, Omega^i)=0 the restriction map to the p-adic completion of de Rham cohomology of every affine open in X is zero. This generalizes an observation of Katz that the restriction map on de Rham cohomology is zero modulo p. We also prove a version of this vanishing for prismatic cohomology, which has consequences for the vanishing in etale cohomology of the generic fiber. We will also discuss the relation of this vanishing to Milnor-Bloch-Kato conjecture, and the obstruction to deducing the above special case of Grothendieck's conjecture from these p-adic results.

This is joint work with Hélène Esnault and Mark Kisin.