Quantisation of extremal Kähler metrics

Abstract

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

v2: 51 pages. Significant re-edit in presentation; version 1 divided into three separate papers, each with new materials (see arXiv:1705.11018 and arXiv:1705.11025). The main theorem and its proof unchanged. v3: updated refereces

References in corpus (2)

• Numerical Calabi-Yau metrics
• Relative stability associated to quantised extremal Kähler metrics

Cited by in corpus (3)

• Geometric quantization of coupled Kähler-Einstein metrics
• Balanced metrics for extremal Kähler metrics and Fano manifolds
• Anticanonically balanced metrics and the Hilbert-Mumford criterion for the ${\delta }_m$-invariant of Fujita-Odaka