2025/01/31 by Pablo Pelaez, Pelaez, Pablo, Reyes, Araceli · 1 citation
Physics and Astronomy · #14C15 #14C17 #14C25 #14C35 #19E15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2501.19147
openalex publication_date 2025/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth projective variety of dimension d over an arbitrary base field k and CHn(X)\mathbb Q be the \mathbb Q-vector space of codimension n algebraic cycles of X modulo rational equivalence, 1≤ n ≤ d. Consider the \mathbb Q-vector subspaces CHn(X)\mathbb Q ⊇ CHnalg(X)\mathbb Q ⊇ CHninc(X)\mathbb Q of algebraic cycles which are, respectively, algebraically and incident (in the sense of Griffiths) equivalent to zero. Our main result computes CHdinc(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 DMk, namely, we show that CHdinc(X)\mathbb Q is given by the second step of the orthogonal filtration F\bullet on CHd(X)\mathbb Q, i.e. F2 CHd (X)\mathbb Q= CHdinc(X)\mathbb Q. The orthogonal filtration F^\bullet on CHn(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. F2 CHnalg(X)\mathbb Q= CHninc(X)\mathbb Q. We also compute in the whole Chow group, CHn(X)\mathbb Q, the second step of the orthogonal filtration F2 CHn(X)\mathbb Q in terms of the vanishing of several intersection pairings.