Quantization of symplectic fibrations and canonical metrics

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