Semiclassical asymptotics for Bergman projections with Gevrey weights

We extend the direct approach to the semiclassical asymptotics for Bergman projections, developed by Deleporte–Hitrik–Sjöstrand for real analytic exponential weights and Hitrik–Stone for smooth exponential weights, to the case of Gevrey weights. We prove that the amplitude of the asymptotic Bergman projection forms a Gevrey symbol whose asymptotic coefficients obey certain Gevrey-type growth rate, and it is constructed by an asymptotic inversion of an explicit Fourier integral operator up to a Gevrey-type small remainder.