The maximal destabilizers for Chow and K-stability

Abstract

Donaldson showed that the constant scalar curvature K\"ahler metrics can be quantized by the balanced Hermitian norms on the spaces of global sections. We explore an analogous problem in the unstable situation. For a K-unstable manifold $(X,L)$, its projective embedding via $\left|kL\right|$ will be Chow-unstable when $k$ is sufficiently large and divisible. There is a unique filtration on $\mathrm{H}^{0}(X,kL)$, that corresponds to the maximal destabilizer for Chow-stability of the embedded variety. On the other hand, there is a maximal destabilizer for K-stability after the work of Xia and Li, which corresponds to the steepest descent direction of K-energy. Based on Boucksom-Jonsson's non-Archimedean pluripotential theory and some idealistic assumptions, we provide a route to show that maximal K-destabilizers are quantized by the maximal Chow-destabilizers.