Anticanonically balanced metrics on Fano manifolds

TL;DR

This paper proves that Fano manifolds with discrete automorphism groups and Kähler-Einstein metrics can be approximated smoothly by anticanonically balanced metrics, using Berezin-Toeplitz quantization to simplify Donaldson's proof.

Contribution

It introduces a simplified proof of the existence of anticanonically balanced metrics converging to Kähler-Einstein metrics on Fano manifolds, leveraging Berezin-Toeplitz quantization.

Findings

Convergence of anticanonically balanced metrics to Kähler-Einstein metrics.

Asymptotic analysis of the convergence rate of Donaldson's iterations.

Application of Berezin-Toeplitz quantization to simplify geometric proofs.

Abstract

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.