Incidence equivalence and the Bloch-Beilinson filtration

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.