Essential dimension via prismatic cohomology

Abstract

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 $h_X^{0,i}$. The proof uses the prismatic cohomology of Bhatt and 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.