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

TL;DR

This paper demonstrates that Kähler-Ricci solitons on Fano manifolds can be approximated by quantized metrics, extending classical methods through spectral gap estimates and quantized invariants.

Contribution

It introduces a new approximation approach for Kähler-Ricci solitons using quantized metrics and Futaki invariants, connecting classical and quantum geometric techniques.

Findings

Approximation of Kähler-Ricci solitons by quantized metrics.

Extension of Donaldson's strategy using spectral gap estimates.

Application to uniqueness and existence results for Fano manifolds.

Abstract

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 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 K\"ahler-Ricci solitons. As corollaries, we recover the uniqueness of K\"ahler-Ricci solitons up to automorphisms, and show how our result also applies to K\"ahler-Einstein Fano manifolds with general automorphism group.