Incidence equivalence and the Bloch-Beilinson filtration
Pablo Pelaez, Araceli Reyes
TL;DR
The paper identifies the incidence-equivalence subgroup $CH^d_{\,\mathrm{inc}}(X)_{\,\mathbb{Q}}$ with the second step $F^2CH^d(X)_{\,\mathbb{Q}}$ of the orthogonal filtration on Chow groups, constructed from Voevodsky's motives. It proves that the exterior and intersection products of algebraically trivial cycles lie in $F^2$, and that over finite or algebraic-closure-of-finite-field bases this identification extends to all codimensions. The authors develop a motivic framework using the slice filtration, motivic cohomology, and base-change arguments to relate Chow groups to $DM_k$, providing evidence that the orthogonal filtration behaves like a Bloch–Beilinson–Murre filtration. They also give an explicit computation of $F^2CH^n(X)_{\,\mathbb{Q}}$ via vanishing intersection pairings and treat key cases, including zero-cycles and the algebraic-closure finite-field setting. Overall, the work offers a concrete motivic description of incidence-equivalence and a robust filtration-theoretic approach to Chow groups compatible with classical conjectures.
Abstract
Let $X$ be a smooth projective variety of dimension $d$ over an arbitrary base field $k$ and $CH^n(X)_{\mathbb Q}$ be the $\mathbb Q$-vector space of codimension $n$ algebraic cycles of $X$ modulo rational equivalence, $1\leq n \leq d$. Consider the $\mathbb Q$-vector subspaces $CH^n(X)_{\mathbb Q} \supseteq CH^n_{\mathrm{alg}}(X)_{\mathbb Q} \supseteq CH^n_{\mathrm{inc}}(X)_{\mathbb Q}$ of algebraic cycles which are, respectively, algebraically and incident (in the sense of Griffiths) equivalent to zero. Our main result computes $CH^d_{\mathrm{inc}}(X)_{\mathbb Q}$ (which coincides with the Albanese kernel $T(X)_{\mathbb Q}$ when $k$ is algebraically closed) in terms of Voevodsky's triangulated category of motives $DM_k$, namely, we show that $CH^d_{\mathrm{inc}}(X)_{\mathbb Q}$ is given by the second step of the orthogonal filtration $F^{\bullet}$ on $CH^d(X)_{\mathbb Q}$, i.e. $F^2 CH^d (X)_{\mathbb Q}= CH^d_{\mathrm{inc}}(X)_{\mathbb Q}$. The orthogonal filtration $F^\bullet$ on $CH^n(X)_{\mathbb Q}$ was introduced by the first author, and is an unconditionally finite filtration satisfying several of the properties of the still conjectural Bloch-Beilinson filtration. We also prove that the exterior product and intersection product of algebraic cycles algebraically equivalent to zero is contained in the second step of the orthogonal filtration. Furthermore, if we assume that the field $k$ is either finite or the algebraic closure of a finite field, then the main result holds in any codimension, i.e. $F^2 CH^n_{\mathrm{alg}}(X)_{\mathbb Q}= CH^n_{\mathrm{inc}}(X)_{\mathbb Q}$. We also compute in the whole Chow group, $CH^n(X)_{\mathbb Q}$, the second step of the orthogonal filtration $F^2 CH^n(X)_{\mathbb Q}$ in terms of the vanishing of several intersection pairings.