Berezin Quantization and Reproducing Kernels on Complex Domains

Let $\Omega$ be a non-compact complex manifold of dimension $n$, $\omega =\partial \overline{\partial }\Psi$ a Kähler form on $\Omega$, and $K_\alpha ( x,\overline{y})$ the reproducing kernel for the Bergman space $A^2_\alpha$ of all analytic functions on $\Omega$ square-integrable against the measure $e^{-\alpha \Psi } |\omega ^n|$. Under the condition

F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on $(\Omega ,\omega )$ which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just $\Omega = \mathbf{C} ^n$ and $\Omega$ a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as $\alpha \to +\infty$. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in $\mathbf{C}^n$. Along the way, we also fix two gaps in Berezin's original paper, and discuss, for $\Omega$ a domain in $\mathbf{C}^n$, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure $|\omega ^n|$.