Analytic Bergman operators in the semiclassical limit

TL;DR

This paper develops approximate semiclassical Bergman projections within analytic microlocal analysis, providing asymptotic expansions and new estimates for Bergman kernels and high tensor powers of line bundles.

Contribution

It introduces a novel approach to semiclassical Bergman projections using analytic microlocal analysis, with explicit asymptotic expansions of their kernels.

Findings

Asymptotic expansion of Bergman kernel functions in the analytic setting

New estimates for Bergman kernels on complex Euclidean spaces

Results applicable to high tensor powers of holomorphic line bundles

Abstract

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.