Asymptotic Chow polystability in Kähler geometry

Details Asymptotic Chow polystability in Kähler geometry Asymptotic Chow polystability in Kähler geometry

2011-05-24

arxiv.org/abs/1105.4773v2

Mathematics

Differential Geometry

Survey paper submitted to the Proceedings of ICCM 2010

Scientific paper

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.

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