Title: Relative stability associated to quantised extremal Kähler metrics,
Abstract: We study algebro-geometric consequences of the quantised extremal Kähler
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--Székelyhidi. In proving them, we further provide an explicit local
density formula for the equivariant Riemann--Roch theorem.