Regular quantizations of Kähler manifolds and constant scalar curvature metrics

In this paper we prove that if a Kähler manifold (M, ω) admits a regular quantization then its scalar curvature is constant. Moreover, we apply this result to the two-dimensional complete Reinhardt domains in C2 to show that such domains admit a regular quantization iff they are biholomorphically isometric to the 2-ball in C2 endowed with the hyperbolic metric.