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Hodge structures through an étale motivic point of view

Ivan Rosas Soto

Abstract

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

Hodge structures through an étale motivic point of view

Abstract

We define the category of étale Chow motives as the étale analogue of Grothendieck motives and proved that it embeds in . This construction provides a characterization of the generalized Hodge conjecture in terms of an étale analogue of it.
Paper Structure (14 sections, 16 theorems, 52 equations)

This paper contains 14 sections, 16 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Étale Chow groups and Lichtenbaum cohomology
  3. Étale Chow groups
  4. Gysin morphism and functoriality properties
  5. Lichtenbaum cohomology groups
  6. Category of étale Chow motives
  7. Correspondences
  8. Operation of correspondences
  9. Action on cycles and cohomology groups
  10. Étale Chow motives
  11. Hodge and Generalized Hodge conjecture
  12. Hodge conjecture and Lichtenbaum cohomology
  13. Generalized Hodge conjecture and Lichtenbaum cohomology
  14. Bardelli's example

Key Result

Proposition 1

Let $X$ be a complex smooth projective variety and consider a sub-Hodge structure $W \subset H^{2k}_B(X,\mathbb{Z}(k))$ of type $(k,k)$. Then $W$ is $L$-algebraic, i.e. $W \subset \text{im}(c^k_L)$, if and only if $W\otimes \mathbb{Q}$ is algebraic.

Theorems & Definitions (39)

  • Proposition : see Proposition \ref{['propE']}
  • Theorem : see Theorem \ref{['teo']}
  • Corollary : see Corollary \ref{['corf']}
  • Remark 2.1.1
  • Proposition 2.1.2
  • proof
  • Remark 2.2.1
  • Proposition 2.2.2
  • proof
  • Definition 2.3.1
  • ...and 29 more