Extremal metrics and K-stability

TL;DR

This paper introduces an algebraic stability criterion for polarized varieties to admit extremal Kähler metrics, extending existing conjectures and providing theoretical and computational support.

Contribution

It generalizes Yau, Tian, and Donaldson's conjectures to extremal Kähler metrics and offers a new algebraic geometric stability criterion with supporting examples.

Findings

Proposes a new stability criterion for extremal Kähler metrics

Provides a GIT-based result motivating the conjecture

Includes example computations supporting the criterion

Abstract

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.