  Given a polarized manifold there are obstructions for asymptotic Chow
semistability described as integral invariants. One of them is an obstruction
to the existence for the first Chern class of the polarization to admit a
constant scalar curvature K\"ahler (cscK) metric. A natural question is whether
or not the other obstructions are linearly dependent on the obstruction to the
existence of a cscK metric. The purpose of this paper is to see that this is
not the case by exhibiting toric Fano threefolds in which these obstructions
span at least two dimension. To see this we show that on toric Fano manifolds
these obstructions are obtained as derivatives of the Hilbert series. This last
observation should be regarded as an extension of the volume minimization of
Martelli, Sparks and Yau.
