Asymptotic Chow polystability in Kähler geometry

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.

Survey paper submitted to the Proceedings of ICCM 2010

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Cited by in corpus (4)

• Cahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
• Quantum moment map and obstructions to the existence of closed Fedosov star products
• Cohomology formula for obstructions to asymptotic Chow semistability
• A note on blow-ups of toric surfaces and cscK metrics