Balanced metrics for K\"ahler-Ricci solitons and quantized Futaki invariants

We show that a Kähler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized Kähler-Ricci solitons. The proof uses a semiclassical estimate on the spectral gap of an equivariant Berezin transform to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized Kähler-Ricci solitons. As corollaries, we recover the uniqueness of Kähler-Ricci solitons up to automorphisms, and show how our result also applies to Kähler-Einstein Fano manifolds with general automorphism group

Spectral aspects of the Berezin transform

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kahler manifolds with constant scalar curvature. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.Comment: Final version, 47 pages. Section on Donaldson's iterations revise

Anticanonically balanced metrics on Fano manifolds

We show that if a Fano manifold has discrete automorphism group and admits a polarized K\"ahler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the K\"ahler-Einstein metric. Our proof is based on a simplification of Donaldson's proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin-Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson's iterations in the anticanonical setting.Comment: 38 page

Balanced metrics for K\"ahler-Ricci solitons and quantized Futaki invariants

We show that a K\"ahler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized K\"ahler-Ricci solitons. The proof uses an equivariant version of Berezin-Toeplitz quantization to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized K\"ahler-Ricci solitons. As a by-product, we show that a K\"ahler-Einstein Fano manifold does not necessarily admit anticanonically balanced metrics in the usual sense when its automorphism group is not discrete.Comment: 46 page

Quantization of symplectic fibrations and canonical metrics

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.Comment: 51 page