# Algebraic Cycles and Motives: On the Splitting of the Bloch–Beilinson Filtration

@article{Beauville2004AlgebraicCA, title={Algebraic Cycles and Motives: On the Splitting of the Bloch–Beilinson Filtration}, author={Arnaud Beauville}, journal={arXiv: Algebraic Geometry}, year={2004} }

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…

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