Quantisation of extremal K\"ahler metrics

TL;DR

This paper demonstrates that extremal Kähler metrics can be approximated by a sequence of metrics satisfying a specific holomorphicity condition related to the Bergman function, bridging geometric analysis and algebraic geometry.

Contribution

It introduces a sequence of Kähler metrics converging to an extremal metric, each satisfying a new holomorphicity condition involving the Bergman function's gradient.

Findings

Sequence of metrics converges to the extremal metric.

Each approximating metric satisfies a holomorphicity condition.

Connects extremal metrics with algebraic quantization methods.

Abstract

Suppose that a polarised K\"ahler manifold $(X,L)$ admits an extremal metric $\omega$. We prove that there exists a sequence of K\"ahler metrics $\{ \omega_k \}_k$, converging to $\omega$ as $k \to \infty$, each of which satisfies the equation $\bar{\partial} \text{grad}^{1,0}_{\omega_k} \rho_k (\omega_k)=0$; the $(1,0)$-part of the gradient of the Bergman function is a holomorphic vector field.