Relative stability associated to quantised extremal K\"ahler metrics

Abstract

We study algebro-geometric consequences of the quantised extremal K\"ahler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa--Sz\'ekelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann--Roch theorem.