Essential dimension via prismatic cohomology

TL;DR

The paper establishes lower bounds on the mod p cohomology of complex varieties using prismatic cohomology, leading to new results on the p-essential dimension and p-incompressibility of certain covers, confirming conjectures for large p.

Contribution

It introduces a novel approach using prismatic cohomology to analyze the essential dimension and p-incompressibility of covers of complex varieties.

Findings

Lower bounds on mod p cohomology dimensions for large p

Proof of p-incompressibility of certain homology covers of abelian varieties

Existence of p-congruence covers that are p-incompressible for Hermitian symmetric domains

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 $h^{0,i}_X$. 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.