  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.
