Analytic Bergman operators in the semiclassical limit

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on ${\mathbb{C}}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.