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-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.

40 pages. Minor corrections and revisions

References in corpus (2)

• Logarithmic adic spaces: some foundational results
• Logarithmic Prismatic Cohomology I