Asymptotics for Bergman projections with smooth weights: a direct approach

TL;DR

This paper extends the direct semiclassical Bergman kernel asymptotics approach, previously used for real analytic weights, to smooth weights, avoiding complex tricks and enabling explicit amplitude construction via Fourier integral operators.

Contribution

It introduces a new method for asymptotic Bergman kernel analysis with smooth weights, bypassing the Kuranishi trick and using explicit Fourier integral operator inversion.

Findings

Successful adaptation of the approach to smooth weights

Explicit construction of the Bergman projection amplitude

Avoidance of the Kuranishi trick in asymptotic analysis

Abstract

We adapt the direct approach to the semiclassical Bergman kernel asymptotics, developed recently by A. Deleporte, J. Sj\"ostrand, and the first-named author for real analytic exponential weights, to the smooth case. Similar to that work, our approach avoids the use of the Kuranishi trick and it allows us to construct the amplitude of the asymptotic Bergman projection by means of an asymptotic inversion of an explicit Fourier integral operator.