Relative uniform $K$-stability over models implies existence of extremal metrics

TL;DR

The paper proves that relative uniform $K$-stability over models guarantees the existence of extremal metrics on polarized smooth complex projective varieties, extending previous results for constant scalar curvature Kähler metrics.

Contribution

It extends the link between $K$-stability and extremal metrics to a broader setting involving relative uniform stability over models.

Findings

Existence of extremal metrics follows from relative uniform $K$-stability over models.

Generalizes Chi Li's result for constant scalar curvature Kähler metrics.

Bridges stability conditions with geometric metric existence in complex geometry.

Abstract

We prove that an extremal metric on a polarised smooth complex projective variety exists if it is $\mathbb{G}$-uniformly $K$-stable relative to the extremal torus over models, extending a result due to Chi Li for constant scalar curvature K\"ahler metrics.