2022/08/22 by Martin, Olivier, Vial, Charles
#14C15 #14C25 #14C30 #14J28 #14J42 #14K05 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.10026
A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step of the Bloch-Beilinson filtration. In this note, we formulate a generalization that allows for rational equivalence of effective zero-cycles of higher degree, at the expense of looking deeper in the Bloch-Beilinson filtration. In the first half, we provide evidence in support of this conjecture in the case of abelian varieties and projective hyper-Kähler manifolds. Notably, we give explicit criteria for rational equivalence of effective zero-cycles on moduli spaces of semistable sheaves on K3 surfaces, generalizing that of Marian-Zhao. In the second half, in an effort to explain our main conjecture, we formulate a second conjecture predicting when the diagonal of a smooth projective variety belongs to a subalgebra of the ring of correspondences generated in low degree.