Uniform Rost nilpotence and birational motives

Abstract

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.