Quantisation of Extremal Kähler Metrics

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.