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Asymptotic Chow polystability in Kähler geometry. (English) Zbl 1246.32026

Ji, Lizhen (ed.) et al., Fifth international congress of Chinese mathematicians. Proceedings of the ICCM ’10, Beijing, China, December 17–22, 2010. Part 1. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-7586-5/pbk; 978-0-8218-7555-1/set). AMS/IP Studies in Advanced Mathematics 51, pt.1, 139-153 (2012).
Summary: 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.
For the entire collection see [Zbl 1235.00045].

MSC:

32Q15 Kähler manifolds
32Q20 Kähler-Einstein manifolds
32Q26 Notions of stability for complex manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
55N91 Equivariant homology and cohomology in algebraic topology
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