(Joint work with Mark Kisin and Alexander Petrov)

Motivated by the instances of Grothendieck’s generalized Hodge conjecture which have a purely algebraic expression, we study the restriction map in cohomology from a smooth projective variety $X$  to an affine $U$. Starting from $X$ being defined over an affine $S$ smooth over $\mathbb Z$,  we consider de Rham cohomology over a dense set of closed points in $S$. Then the restriction is ‘controlled’ by the global differential forms on $X$. When we go $p$-adically we prove the ’same’ result with values in a certain separated quotient of Bhatt-Scholze prismatic cohomology which maps to the separated quotient of derived $p$-complete de Rham cohomology. 

We construct examples which show that one needs an extra assumption to lift this control to the whole cohomology.

Caro-D’Addezio showed that for derived  $p$-complete de Rham cohomology modulo $p$-torsion it is controlled by $H^1$ of differential forms in addition. We lift this to prismatic cohomology modulo $I$-torsion,  which also has a corollary on étale cohomology of the rigid analytic fibre.