Let $k$ be an algebraically closed field of characteristic $p>0$ and write $W:=W(k)$ for its ring of Witt vectors. Consider a smooth cubic fourfold $X_{0}\subset\mathbb{P}^{5}_{k}$ and let $F(X_{0})$ be its Fano scheme of lines. Then the incidence correspondence \begin{equation*} F(X_{0})\leftarrow I(X_{0})\subset X_{0}\times_{k} F(X_{0})\rightarrow X_{0} \end{equation*} induces a morphism on cohomology \begin{equation*} H_{\text{cris}}^{4}(X_{0}/W)\rightarrow H_{\text{cris}}^{2}(F(X_{0})/W) \end{equation*} Does $a$ induce an isomorphism on the primitive crystalline cohomology?
Motivation:-
For primes $\ell\neq p$, consider the morphism \begin{equation*} H_{\text{ét}}^{4}(X_{0},\mathbb{Z}_{\ell})_{\text{prim}}\rightarrow H_{\text{ét}}^{2}(F(X_{0}),\mathbb{Z}_{\ell})_{\text{prim}} \end{equation*} induced by the incidence correspondence. This is an isomorphism because $X_{0}$ lifts to a cubic hypersurface $X/W$, and then we use the smooth proper base change theorem to reduce to considering the corresponding morphism over $\mathbb{C}$, which is an isomorphism by Beauville and Donagi's paper ``La variete des droites d'une hypersurface cubique de dimension 4''.
