# Relative stability associated to quantised extremal K\"ahler metrics

## TL;DR

This paper explores the implications of quantised extremal K"ahler metrics, establishing their connection to stability notions in algebraic geometry and providing new proofs and formulas related to extremal manifolds.

## Contribution

It demonstrates that quantised extremal K"ahler metrics imply weak relative Chow polystability and offers an alternative proof of K-semistability for extremal manifolds.

## Key findings

- Quantised extremal metrics imply weak relative Chow polystability.
- Asymptotic weak relative Chow polystability and K-semistability are derived from these metrics.
- Provides an explicit local density formula for the equivariant Riemann--Roch theorem.

## 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.