  A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism
from the rational Chow ring of an abelian variety to its rational Chow ring
modulo numerical equivalence admits a (canonical) section. Motivated by
Beauville's splitting principle, we formulate a conjectural Section Property
which predicts that for smooth projective holomorphic symplectic varieties
there exists such a section of algebra whose image contains all the Chern
classes of the variety. In this paper, we investigate this property for (not
necessarily symplectic) varieties with Chow motive of abelian type. We
introduce the notion of symmetrically distinguished abelian motive and use it
to provide a sufficient condition for a smooth projective variety to admit such
a section. We then give series of examples of varieties for which our theory
works. For instance, we prove the existence of such a section for arbitrary
products of varieties with Chow groups of finite rank, abelian varieties,
hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on
an abelian surface or a Kummer surface or a K3 surface with Picard number at
least 19, and generalized Kummer varieties. The latter cases provide evidence
for the conjectural Section Property and exemplify the mantra that the motives
of holomorphic symplectic varieties should behave as the motives of abelian
varieties, as algebra objects.
