Asymptotic Chow polystability in K\"ahler geometry
  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. 
Akito Futaki
math.DG