GEOMETRIC QUANTIZATION OF COUPLED KAHLER-EINSTEIN METRICS

We study the quantization of coupled Kahler-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 DonaldsonFutaki 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. 1. Introduction 1817 2. Preliminaries 1821 3. Geometric quantization 1827 4. Existence and weak convergence of balanced metrics 1838 5. Towards the C1-convergence 1840 Acknowledgments 1847 References 1847