Quantization of symplectic fibrations and canonical metrics

TL;DR

This paper explores the relationship between Berezin-Toeplitz quantization, symplectic fibrations, and the geometry of vector bundles, providing new insights into spectral gaps, convergence rates, and balanced metrics in complex geometry.

Contribution

It introduces a novel connection between quantization of symplectic fibrations and the analysis of vector bundles, including refined estimates for convergence to balanced metrics.

Findings

Spectral gap estimates for Berezin transform

Convergence rates of Donaldson's iterations

Refined scalar case estimates for Kähler manifolds

Abstract

We relate Berezin-Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson's iterations towards balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson's iterations towards balanced metrics on K\"ahler manifolds with constant scalar curvature.