We study the quantization of coupled Kähler–Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, called “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 an algebrogeometric 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.

我们研究耦合 Kähler-Einstein (CKE) 度量的量化，即我们通过规范的 Bergman 度量来近似 CKE 度量，称为“平衡度量”。我们证明了负第一陈类平衡度量的存在和弱收敛，而对于正第一陈类，我们引入了代数几何障碍，它在 Donaldson-Futaki 不变量和 Chow 权重之间进行插值。然后我们展示了在这个障碍消失的情况下，CKE 流形上平衡度量的存在和弱收敛。此外，限于自同构群离散的情况，我们还讨论了近似解和梯度流方法，以实现平滑收敛。