  It is conjectured that the existence of constant scalar curvature K\"ahler
metrics will be equivalent to K-stability, or K-polystability depending on
terminology (Yau-Tian-Donaldson conjecture).
  There is another GIT stability condition, called the asymptotic Chow
polystability.
  This condition implies the existence of balanced metrics for polarized
manifolds $(M, L^k)$ for all large $k$.
  It is expected that the balanced metrics converge to a constant scalar
curvature metric as $k$ tends to infinity under further suitable stability
conditions. In this survey article I will report on recent results saying that
the asymptotic Chow polystability does not hold for certain constant scalar
curvature K\"ahler manifolds. We also compare a paper of Ono with that of Della
Vedova and Zuddas.
