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

TL;DR

The paper demonstrates that if certain realization functors are conservative, then motivic zeta functions are rational and numerical motives are Kimura-finite, leading to a Tannakian structure of the category of numerical motives.

Contribution

It establishes a link between the conservativity of realization functors and the rationality of motivic zeta functions and Kimura-finiteness of numerical motives, under recent cohomological assumptions.

Findings

Motivic zeta functions are rational under conservative realizations.

Numerical motives are Kimura-finite if realization functors are conservative.

The category of numerical motives is essentially Tannakian.

Abstract

We prove: if the (\'etale or de Rham) realization functor is conservative on the category $DM_{gm}$ 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 $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives coming from generic hyperplane sections of smooth affine varieties.