Asymptotic Chow polystability in Kähler geometry

Abstract

Abstract. It is conjectured that the existence of constant scalar curvature Kähler 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 Kähler manifolds. We also compare a paper of Ono with that of Della Vedova and Zuddas.