  The holomorphic invariants introduced by Futaki as obstruction to the
asymptotic Chow semistability are studied by an algebraic-geometric point of
view and are shown to be the Mumford weights of suitable line bundles on the
Hilbert scheme. These invariants are calculated in two special cases. The first
is a projective bundle over a curve of genus at least two, and it is shown that
it is asymptotically Chow polystable (with every polarization) if and only the
underlying vector bundle is slope polystable. This proves a conjecture of
Morrison with the extra assumption that the involved polarization is
sufficiently divisible. Moreover it implies that a projective bundle is
asymptotically Chow polystable (with every polarization) if and only if it
admits a constant scalar curvature Kaehler metric. The second case is a
manifold blown-up at points, and new examples of asymptotically Chow unstable
constant scalar curvature Kaehler classes are given.
