On the splitting of the Bloch-Beilinson filtration

TL;DR

This paper explores the conditions under which the conjectural Bloch-Beilinson filtration on the Chow ring of algebraic cycles splits, with evidence and examples related to K3 surfaces, abelian varieties, Calabi-Yau threefolds, and symplectic manifolds.

Contribution

It investigates the splitting of the Bloch-Beilinson filtration, providing examples, counterexamples, and conjectural insights for various classes of algebraic varieties.

Findings

Splitting implies injectivity of divisor class subalgebra into cohomology.

Calabi-Yau threefolds do not satisfy the splitting property.

Conjecture that symplectic manifolds do satisfy the splitting property.

Abstract

For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Namely, for a regular variety X, it implies that the sub-Q-algebra of CH(X) spanned by divisor classes injects into the cohomology of X . We give examples of Calabi-Yau threefolds which do not satisfy this property. On the other hand we conjecture that the property does indeed hold for (holomorphic) symplectic manifolds, and we give some (weak) evidence in favour of this conjecture.