  For $X$ a smooth, proper complex variety we show that for $p\gg 0$, the
restriction of the mod $p$ cohomology $H^i(X,\mathbb{F}_p)$ to any Zariski open
has dimension at least $h^{0,i}_X$. The proof uses the prismatic cohomology of
Bhatt-Scholze.
  We use this result to obtain lower bounds on the $p$-essential dimension of
covers of complex varieties. For example, we prove the $p$-incompressibility of
the mod $p$ homology cover of an abelian variety, confirming a conjecture of
Brosnan for sufficiently large $p.$ By combining these techniques with the
theory of toroidal compactifications of Shimura varieties, we show that for any
Hermitian symmetric domain $X,$ there exist $p$-congruence covers that are
$p$-incompressible.
