Effective zero-cycles and the Bloch-Beilinson filtration

TL;DR

This paper explores a generalization of Voisin's conjecture relating rational equivalence of zero-cycles to the Bloch-Beilinson filtration, providing evidence for specific varieties and proposing a new conjecture about the diagonal in the ring of correspondences.

Contribution

It formulates a broader conjecture on zero-cycles and the Bloch-Beilinson filtration, with explicit criteria for certain moduli spaces and a new conjecture on the diagonal in the correspondence ring.

Findings

Evidence supporting the conjecture for abelian varieties and hyper-Kähler manifolds.

Explicit criteria for rational equivalence on moduli spaces of sheaves on K3 surfaces.

A new conjecture on the diagonal's position in the algebra of correspondences.

Abstract

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-K\"ahler 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.