Conservativity of realizations implies that numerical motives are Kimura-finite and motivic zeta functions are rational

Abstract

We prove: if the (étale or de Rham) realization functor is conservative on the category $D{M}_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $D{M}_{gm}(k)$ whenever $\mathrm{char}k=0$. We apply this statement to exterior powers of motives coming from generic hyperplane sections of smooth affine varieties.

A collection of minor corrections made

References in corpus (1)

• Detecting effectivity of motives, their weights, connectivity, and dimension via Chow-weight (co)homology: a "mixed motivic decomposition of the diagonal"