An example of asymptotically Chow unstable manifolds with constant scalar curvature

TL;DR

This paper provides a counterexample showing that the previously established link between constant scalar curvature Kähler metrics and asymptotic Chow stability does not hold when the automorphism group is not discrete.

Contribution

The paper constructs an example of a polarized manifold with constant scalar curvature that is asymptotically Chow unstable when its automorphism group is not discrete.

Findings

Counterexample to Donaldson's stability result

Demonstrates failure of asymptotic Chow stability in non-discrete automorphism cases

Highlights limitations of stability criteria in Kähler geometry

Abstract

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature K\"ahler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.