# Geometric quantization of coupled K\"ahler-Einstein metrics

## TL;DR

This paper investigates the approximation of coupled Kähler-Einstein metrics using balanced metrics, establishing existence, convergence, and obstructions, with special focus on cases with discrete automorphism groups.

## Contribution

It introduces a new algebro-geometric obstruction for positive first Chern class and proves convergence of balanced metrics under certain conditions.

## Key findings

- Existence of balanced metrics for negative first Chern class
- Introduction of an obstruction for positive first Chern class
- Weak convergence of balanced metrics on CKE manifolds

## Abstract

We study the quantization of coupled K\"ahler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.