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Some remarks on Chow correspondences

Pablo Pelaez

TL;DR

The paper characterizes when a Chow correspondence $\Lambda$ induces the zero map on the Albanese level by translating the problem into the motivic setting of $DM_k$ and the orthogonal filtration. It introduces a criterion involving the functor of points of the Albanese variety $\mathcal{A}$ and its orthogonal cover $bc_{\leq -1}$, linking the vanishing of $\psi$ to a lift through $bc_{\leq -1}(\mathcal{A})$ and to the zero–motive component in the orthogonal filtration. Moreover, it shows that the orthogonal filtration detects square-equivalence to zero and the $H^{*2}$ step of H. Saito's filtration, with stronger conclusions available over $\mathbb{C}$ under appropriate hypotheses. The results provide a motivic perspective on a classical question about algebraic cycles, Albanese kernels, and reg­ular homomorphisms, and they illuminate how filtrations in Voevodsky’s framework control geometric equivalence relations on Chow groups. Overall, the work ties birational and cohomological refinements to concrete cycle-theoretic criteria via motivic functors and the Albanese formalism.

Abstract

We study, in the context of Voevodsky's triangulated category of motives, several adequate equivalence relations (in the sense of Samuel) on the graded Chow ring $CH^\ast (X\times Y)$ for $X$, $Y$ smooth projective varieties over a field.

Some remarks on Chow correspondences

TL;DR

The paper characterizes when a Chow correspondence induces the zero map on the Albanese level by translating the problem into the motivic setting of and the orthogonal filtration. It introduces a criterion involving the functor of points of the Albanese variety and its orthogonal cover , linking the vanishing of to a lift through and to the zero–motive component in the orthogonal filtration. Moreover, it shows that the orthogonal filtration detects square-equivalence to zero and the step of H. Saito's filtration, with stronger conclusions available over under appropriate hypotheses. The results provide a motivic perspective on a classical question about algebraic cycles, Albanese kernels, and reg­ular homomorphisms, and they illuminate how filtrations in Voevodsky’s framework control geometric equivalence relations on Chow groups. Overall, the work ties birational and cohomological refinements to concrete cycle-theoretic criteria via motivic functors and the Albanese formalism.

Abstract

We study, in the context of Voevodsky's triangulated category of motives, several adequate equivalence relations (in the sense of Samuel) on the graded Chow ring for , smooth projective varieties over a field.
Paper Structure (12 sections, 4 theorems, 26 equations)

This paper contains 12 sections, 4 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Notation
  4. Voevodsky's triangulated category of motives
  5. Voevodsky's slice filtration
  6. The orthogonal filtration
  7. Adequate equivalence relations and Voevodsky's motives
  8. Adequate equivalence relations
  9. Product of equivalence relations MR1189891*1.3
  10. H. Saito's filtration MR1189891*Ex. 1.8
  11. Regular homomorphisms
  12. The orthogonal filtration for the functor of points of an abelian variety

Key Result

Theorem 4.1.4

With the notation and conditions of ss.results and diag.mot-alb-diag.mainthm. There exist a map $\Lambda ' : M(Y)(-n)[-2n]\rightarrow bc _{\leq -1}(\mathcal{A})$ in $DM_{k}$ such that diag.mainthm commutes if and only if the following condition holds:

Theorems & Definitions (10)

  • Theorem 4.1.4
  • proof
  • Remark 4.2.7
  • Proposition 4.2.8
  • proof
  • Theorem 4.3.1
  • proof
  • Theorem 4.3.2
  • proof
  • Remark 4.3.3