Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic cycles of some Fano varieties with Hodge structure of level one

Pedro Montero, Iván Rosas-Soto

Abstract

We study Chow groups and étale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We extend these results to an étale motivic context and recover an analogous finite-dimensionality in the sense of Kimura. We further analyse algebraic cycles on other smooth Fano manifolds with Hodge structures of level one and, as an application, we prove the integral Hodge conjecture for smooth quartic double fivefolds by means of the étale motivic approach.

Algebraic cycles of some Fano varieties with Hodge structure of level one

Abstract

We study Chow groups and étale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We extend these results to an étale motivic context and recover an analogous finite-dimensionality in the sense of Kimura. We further analyse algebraic cycles on other smooth Fano manifolds with Hodge structures of level one and, as an application, we prove the integral Hodge conjecture for smooth quartic double fivefolds by means of the étale motivic approach.

Paper Structure

This paper contains 19 sections, 38 theorems, 201 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on Chow groups
  3. Preliminaries on étale Chow groups
  4. Étale Chow groups
  5. Étale motivic cohomology of complete intersections
  6. Chow groups of complete intersections with Hodge level one
  7. Chow groups of an intersection of two quadrics
  8. Chow groups of an intersection of three quadrics
  9. Case when
  10. Chow groups of when
  11. Étale Chow groups of complete intersection with Hodge level one
  12. Fano threefolds
  13. Cubic fivefolds
  14. Intersections of two and three quadrics
  15. Decomposition of finite dimensional motives
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $f:X\to \mathbb{P}^5$ be a smooth double cover branched along a smooth quartic hypersurface $B\in |\mathcal{O}_{\mathbb{P}^5}(4)|$. Then the integral Hodge conjecture holds for $X$.

Theorems & Definitions (95)

  • Theorem 1.1: Theorem \ref{['thm:Hodge']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: VoiB
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 3.1
  • ...and 85 more