  We introduce a homological Lefschetz conjecture on (rational) Chow groups,
which can be deduced from some well known conjectures, and illustrate it by a
series of key examples. We then prove the injectivity of the push-forward
morphism on Chow groups, induced by the closed embedding of the Theta divisor
in it's Jacobian $J(C)$. Here $C$ is a smooth irreducible complex projective
curve.
