Artin vanishing along the $\ell$-adic tower of an abelian variety

Abstract

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible ${\mathbb{F}}_{\ell }$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[{\ell }^e{]}^{\ast }:{\mathrm{H}}^i(A,[{\ell }^n{]}^{\ast }F)\to {\mathrm{H}}^i(A,[{\ell }^{n+e}{]}^{\ast }F)$ is zero for all $n\ge 0$ and all $i>\dim \mathrm{Supp}F$. In particular $\underset{n}{\underrightarrow{\lim }}{\mathrm{H}}^i(A,[{\ell }^n{]}^{\ast }F)=0$ for $i>\dim \mathrm{Supp}F$, which answers a question of Bhatt--Schnell--Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier--Mellin transform of a perverse sheaf. With $\ell$-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault--Kerz. The corresponding estimates for the completed transform with ${\mathbb{F}}_{\ell }$-coefficients are new and hold even when Hard Lefschetz fails.

arXiv comment29 pp, 1 figure