发表机构
Boston University; University of Notre Dame（波士顿大学; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明阿贝尔簇上可构造$\mathbb{F}_{\ell}$-层沿$\ell$-adic塔的Artin消没，回答Bhatt--Schnell--Scholze问题，并给出Fourier--Mellin变换上同调层的余维数估计，推广Esnault--Kerz结果。
AI 中文摘要
设$A$是代数闭域上的$g$维阿贝尔簇，且$\ell$是该域中可逆的素数。对于$A$上的任意可构造$\mathbb{F}_{\ell}$-层$F$，我们证明存在一个依赖于$F$的整数$e$，使得对所有$n\geq0$及所有$i>\dim\operatorname{Supp}F$，拉回映射$[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$为零。特别地，当$i>\dim\operatorname{Supp}F$时，$\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$，这回答了Bhatt--Schnell--Scholze的问题。我们的方法还给出了perverse层的Fourier--Mellin变换的上同调层的支撑集的尖锐余维数估计。对于$\ell$-adic系数，我们移除了Esnault--Kerz估计中的算术性假设。对于$\mathbb{F}_{\ell}$系数的完备变换的相应估计是新的，并且在Hard Lefschetz失效时仍然成立。
英文摘要
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 $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$ is zero for all $n\geq0$ and all $i>\dim\operatorname{Supp}F$. In particular $\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$ for $i>\dim\operatorname{Supp}F$, 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.
Comments29 pp, 1 figure
