Hodge structures through an \'etale motivic point of view

TL;DR

This paper introduces the category of étale Chow motives as an étale analogue of Grothendieck motives, embedding it into a motivic derived category and relating it to the generalized Hodge conjecture.

Contribution

It defines étale Chow motives and demonstrates their embedding into a motivic category, offering a new perspective on the generalized Hodge conjecture.

Findings

Étale Chow motives form a new category analogous to classical motives.

Embedding of étale Chow motives into the étale motivic derived category.

Characterization of the generalized Hodge conjecture via étale motives.

Abstract

We define the category of \'etale Chow motives as the \'etale analogue of Grothendieck motives and proved that it embeds in $\text{DM}_{\text{\'et}}(k)$. This construction provides a characterization of the generalized Hodge conjecture in terms of an \'etale analogue of it.