Uniform Rost 幂零性与双有理 motive
Uniform Rost nilpotence and birational motives
中文总结 AI 辅助
本文为基变换理想建立显式幂零性界，证明有效泛下降，改进均匀界，并推广检测子至理想层面，应用于三维簇。
中文摘要 AI 辅助
对于域扩张 $E/k$ 和 Chow motive $M$，令 $I_E(M)=\ker\bigl(\operatorname{End}_k(M)\longrightarrow \operatorname{End}_E(M_E)\bigr)$ 为基变换理想。我们在若干几何情形中建立了这些理想的显式幂零性界。我们还证明了有效的泛下降：若 $M$ 是 $h(X)(a)$ 的一个直和项且 $M_{k(X)}$ 具有均匀 Rost 指数 $s$，则 $M$ 具有指数 $s(\dim X+1)$。这给出了维数为 $2n-2$ 的扭曲 Milnor 超平面截线的积分指数 $2n-1$。在特征零情形，多重线性 Rost 滤过的加权局部块细化改进了 Gille 的均匀界，并将 Rosenschon--Sawant 的逐元素估计提升为理想层面的均匀界。利用 Kahn--Sujatha 对纯双有理 motive 的描述，我们获得了从双有理 motive 到普通 Chow motive 的定量提升结果，并应用于三维簇，包括与环面模型双有理的簇以及允许对角线分解支撑在曲面上的簇。最后，我们将 Kok--Zhou 检测子从单个对应推广到整个基变换理想，通过临界细化非分歧上同调群的均匀消没获得显式的理想幂零性界。
英文摘要
For a field extension $E/k$ and a Chow motive $M$, let $I_E(M)=\ker\bigl(\operatorname{End}_k(M)\longrightarrow \operatorname{End}_E(M_E)\bigr)$ be the base-change ideal. We establish explicit nilpotence bounds for these ideals in several geometric settings. We also prove effective generic descent: if $M$ is a summand of $h(X)(a)$ and $M_{k(X)}$ has uniform Rost exponent $s$, then $M$ has exponent $s(\dim X+1)$. This yields the integral exponent $2n-1$ for twisted Milnor hyperplane sections of dimension $2n-2$. In characteristic zero, a weighted local-block refinement of the multilinear Rost filtration improves the uniform bounds of Gille and upgrades the elementwise estimates of Rosenschon--Sawant to uniform bounds at the ideal level. Using the Kahn--Sujatha description of pure birational motives, we obtain quantitative lifting results from birational to ordinary Chow motives, with applications to threefolds, including varieties birational to toric models and varieties admitting a decomposition of the diagonal supported on a surface. Finally, we extend the Kok--Zhou detector from individual correspondences to entire base-change ideals, obtaining an explicit ideal nilpotence bound from uniform annihilation of the critical refined unramified cohomology groups.
