The semiclassical analysis of Bergman kernels is a multifaceted subject, with applications to physics, complex analysis, and geometry. Locally, the study of Bergman kernels can often be reduced to the analysis of the orthogonal projection onto an exponentially weighted space of holomorphic functions on some complex domain from the ambient weighted L^2-space, and in this talk, we shall be concerned with the semiclassical asymptotics for Bergman kernels in this setting. We shall discuss a direct approach to the construction of asymptotic Bergman projections, developed with A. Deleporte and J. Sjöstrand in the case of real analytic exponential weights, and with M. Stone in the case of smooth weights. The direct approach avoids the use of the Kuranishi trick and allows us, in particular, to give a simple proof of a result due to O. Rouby, J. Sjöstrand, S. Vu Ngoc, and to A. Deleporte, stating that, in the analytic case, the Bergman projection can be described up to an exponentially small error.

https://indico.math.cnrs.fr/event/10613/